Just to add to the good advice Mike has alread given...

There are studios out there that do not make what most would consider "violent" games. Obviously, Nintendo's internal development teams don't do a lot of extreme bloody violence (Mario, Pokemon, Zelda, etc.). Microsoft has studios that work more on fantasy based games as well as racing games. And, of course, there are studios working on sports titles, flight sims, and other not-so-gory games.

One such studio is the one I work at, Amaze Entertainment. We focus primarily on licensed titles that usually hit a Teen ESRB rating or below. That pretty much means zero blood and gore is involved, and any violence that exists is of a very cartoony and stylized nature (think Disney and Dreamworks animation).

Keep in mind that the game industry is not all about Doom and Quake and Halo and GTA. Do a search on Gamasutra, and you'll find a range of studios offering work in different genres of games.