In the beginning of the manga, Luffy claimed that he wanted at least ten people to join his crew before entering the Grand Line (not counting himself). This was changed after Usopp told Luffy that after "the chef" joined them, they would rescue Nami and go to the Grand Line.
Although this is long past the expiration date, it can still serve as evidence that there will be at least ten Straw Hats at one point of the manga (excluding Luffy himself). Luffy also claimed in the beginning of the manga, that he would gather a crew strong enough to surpass his childhood hero Red-Haired Shanks.
Usually when he asks someone to join, Luffy is turned down completely; though the way of which he is turned down varies. Luffy never forces people to join his crew against their will. When Luffy asked Brook during the Thriller Bark Arc, Brook said yes immediately. Luffy did not directly ask Usopp to join the Straw Hats, however when Luffy, Zoro and Nami were about to sail off, Luffy assumed that Usopp would join them as they had just fought off Kuro. The only exception was Franky who was stated to be too stubborn to ever accept joining Luffy willingly (his swim briefs had to be stolen just to lead him back to the Straw Hat Pirates), but ends up joining willingly anyway after being moved by Iceburg's words.
However if the person really interests Luffy, he will still try to get them to join anyway regardless of their initial decision. While most people turn down Luffy's offer regardless of how hard he tries, only those convinced enough will join for reasons of their own. However, even if he is rejected, Luffy will treat the person as if they were already a crew member, much to their annoyance. He'll sometimes act as if it is his decision and not theirs whether they join or not (though it is worth noting that when Luffy's decided someone is joining they do eventually join).
The major exceptions to this rule are Usopp, who was thrilled to be asked to join and accepted immediately, and Nico Robin, who invited herself into the crew and had to convince the crew to let her stay. Also notable was Brook, who actually accepted Luffy's initial invitation to join the crew but later claimed that he could not honor it due to his lack of a shadow, which would result in his death should he step into sunlight, as well as unfinished business with the zombies of Thriller Bark. However, after the Straw Hats and Brook finally defeated the zombies, Brook later asked if they would honor his initial acceptance and Luffy gladly welcomed him into the crew, though everyone else were shocked by Brook just asking like that. Jinbe also wanted to join after being asked but stated he had unfinished business to do, but did state if they still want him to join after that's done, he will gladly join the Straw Hats.
Luffy does not seem to mind if someone he wants to recruit has a profession or not. He allowed Usopp to join, and later on names him the team's sniper, and considered Vivi and Carue members even though her only "profession" would be princess. He also did not know that Chopper was a doctor and wanted him to join for his seven transformation abilities. He also offered Gaimon and several Thriller Barkzombies the chance to join his crew, when they had no apparent usefulness to the crew aside from experience and their uniqueness respectively. Also, when a crew member finally joins, they are often run out of their home by loved ones.
Though most adventures in One Piece are random, there seems to be a 'process' for recruitment:
First, a situation arises where someone new is needed. The crew comes across someone who fulfills the criteria (i.e. Chopper the doctor, Nami the navigator, Franky the shipwright, Sanji the cook etc.). Luffy offers the person a position, and is usually turned down initially (which usually causes Luffy to comically "refuse this refusal").
Meanwhile, a situation arises where, some way or another, Luffy ends up battling a 'boss' character, and the would-be crew mate battles an underling (i.e. Luffy vs Don Krieg after Sanji battles Gin and Pearl). The would-be crewmate typically defeats one of the villain's henchmen, or otherwise plays a crucial role in the Straw Hats' victory (for example, Brook defeats Tararan and reveals the zombies' weakness, despite losing to Ryuma).
During the arc in which the new crewmate is introduced their backstory and dreams are revealed, (although Nami and Robin's back stories were revealed later) thus convincing them to join the Straw Hat Crew.
Meanwhile, Luffy's current crew would assist him in battling various enemies generally in rank of abilities and overall strength which is matched by whoever they are battling (i.e. Zoro vs Kaku, Sanji vs Jabra, Chopper vs Kumadori, Nami vs Kalifa), the enemies fought also have similar fighting styles (Zoro vs Hachi who are both swordsman, Sanji vs Kuroobi who are both melee fighting experts, and Usopp vs Chew who are both ranged weaponry experts), this is more consistent when it concerns Sanji and Zoro.
This process also led many fans to think Vivi would become a permanent crew member.
The only crew members not to follow this process were the female members, Nami and Robin, both of whom never helped fight an enemy during their joining and had their backstories fully revealed later on, despite initially being hinted at. Nami's official recruitment was when Luffy, Zoro, Usopp and Sanji defeated Arlong and his pirates. Robin was the first to initially willingly join, and officially joined after Luffy's crew came to her rescue from CP9.
As a side note to this process, Luffy has several times stated the need for a musician, because he believes that "pirates love music" due to him being influenced by Shanks the captain of the Red-Haired Pirates whom are mostly seen singing and partying. As a result, whenever the need for a new crew member arises, Luffy would usually bring up the musician position despite it being completely unrelated to the problem at hand. While Franky plays the ukulele in addition of being a shipwright, Luffy's plea was finally fulfilled when Brook joined the crew. There are still lots of roles on the ship that need to be filled like having a permanent helmsman.
Luffy is the captain as he founded the crew.
The numbers given to each crew member are:
Monkey D. Luffy (Captain)
Roronoa Zoro (Swordsman)
Tony Tony Chopper (Doctor)
Nico Robin (Archaeologist)
Usually when numbering the crew on color spreads however, Oda will include Luffy as #1 thus the crew members get +1 added onto their numbers. For most of the crew members, Oda designates their number in the title of the chapter in which they join. For some like Nami and Usopp, their numbers were designated after they returned to the crew for previously leaving it. Only Chopper and Robin so far have no chapters whose titles designate them. In the case of Robin, though she has yet to have a number chapter, she however was numbered as the 7th one in the anime. This however refers to her being the 7th character in the crew which includes Luffy.
If a Straw Hat crew member leaves the crew without Luffy's permission, Luffy does his best to recruit them back. So far, Nami, Robin, and Usopp (and Chopper through the Davy Back Fight, against his will) have all left the crew, and all of them have returned. Nami rejoined after Arlong was defeated, who blackmailed her into betraying the crew, while Robin rejoined the crew after they convinced her that they would stand by her even if it meant declaring war on the entire world, and saving her from Enies Lobby.
While Robin and Nami were never on bad terms with the crew when they left, any suspicion that they had betrayed the Straw Hats was soon proven false. Usopp was a different case though. He had officially left the Straw Hat crew, over a dispute concerning the Going Merry (yet returned to the crew in his alias "Sogeking" to help in the rescue of Nico Robin) and his own insecurities about his strength. Afterward, Usopp originally planned to make the crew beg for him to return, but Zoro had given the crew a stern speech about him rejoining them on their terms, not his, and would accept him only if Usopp begged them after a thorough apology. After being ignored while they were fighting off Garp, Usopp finally apologized, and Luffy happily and tearfully accepted him back into the crew.