Many people like "Condo" living in Los Angeles because condo developments for the most part are secure. Some have 24 hour security with gated entrance and parking. This would be a first choice in my opinion. Neighborhoods are a "patchwork" in Los Angeles so it's impossible say which is safer, you might want to contact the local police department to get more information on that. On a "budget" you should look in the San Fernando Valley area for available property.
Contact a Realtor who works in that area and they will be able to guide you.Not all Realtors work with Rental so be sure to inquire about that.
Condos in this area are selling for a decent price & if anyone has their places up for rent they should be renting for an affordable amount. Also check in the areas of Silver Lake (a little pricier), Melrose Hill, Echo Park, these are all areas that are in a nice commuting distance to downtown LA & also easy freeway access to get to Burbank.
I'm bringing up these areas because my buyer for this condo works in Burbank but also needs to travel to downtown LA & UCLA medical center. She likes this location as it gets her to all of these places with some ease.
When looking for rentals & searching online click on my profile here on Trulia & read my 2 blog posts called "RED FLAGS - RENTAL SCAMS - PUBLIC BEWARE" this will also aid you on your quest for a god rental property.
I don't look back on this same Trulia posting for answers after mine.
Emily S. Knell
Realtor Since 1996
Realty ONE Group
But LA in general is experiencing lots of gentrification and urban inflow... so the situation changes drastically from one block to the next or one side of the fwy to the other....
You really have to come over and drive around at various times to get a sense of what is comfortable for you.