Statistically the crime rate in the last few years has been about average with the rest of the country in Los Angeles (as opposed to 5 years ago when it was actually pretty high). Like any other city though, you should just use common sense. Don't go down dark alleys in Hollywood, etc.
Generally the most dangerous areas in LA are to the south of most tourist locations in places like Compton and Inglewood (where there is a lot of gang activity). There is a fair amount of crime in Hollywood as well, but you should be okay so long as you don't make yourself an easy mark.
In Hollywood specifically, in the last 6 months there have been: 1 murder 19 rapes 125 assaults 224 robberies
It is the 31st most dangerous neighborhood in Los Angeles (out of 209 neighborhoods). (All this is according to the LA Times Neighborhood mapping project--a great resource finding this kind of thing out in the LA area.)
Don't let that scare you too much, but do use street smarts. Enjoy your stay. LA is a lot of fun.
I lived in Hollywood for the first 6 months when I moved to LA. In general if you stay in well lit areas at night and don't venture too far off the main blvd.'s like Hollywood, Sunset, Vine, etc then you should be fine. Just be aware of your surroundings and keep valuables out of sight. Criminals are always looking for an easy target. If you have to talk on a cell phone it is best to stop, if you are walking on the side walk, and lean back against a wall or fence so you can see who is around you. The further east you go in Hollywood the more undesirable it gets. There are a lot of homeless people asking for spare change etc. so if you are in the habit of helping out just make sure you carry some change in a pocket. Do not pull out your wallet or purse to give money or you may end up "giving" more than you wanted to. There are some great sights to see in and around Hollywood. The main tourist area is around Hollywood blvd and Highland Ave, which is where the Kodak theater is. There is also the Wax museum, The Chinese theater, El Capitan theater and several other tourist attractions. Be sure to take a trip up to the Observatory in Griffith park. Be safe and enjoy your trip.