What is your budget? that would help but as you know everywhere is expensive in LA now days.
Here are my suggestions:
If I was still young and single, I would search in the area along Wilshire Blvd and just west of Highland and up to La Cienega Blvd. There's a mix of small older beautiful Spanish style apartments and newer more luxury ones. Personally, I would try to find one that has a washer/dryer in the unit. This would save so much time on keeping up with your laundry and maybe some change since laundromats are charging a lot to wash clothes.
Wilshire Blvd just west of Highland
Another area you mentioned was Crenshaw, maybe Leimert Park neighborhood. There as some nice Spanish style apartment in this neighborhood, and it's close and walkable to the K line, but if you're taking the train to work you would have to transfer to the E Line to head towards downtown and into Boyle Heights. Driving in the morning would be pretty bad heading towards downtown/Boyle Heights.
Typical looking apartments in Leimert Park, although they have built some luxury apartments along Crenshaw along with the train runs
North Hollywood specially the area around the subway is walkable and full of cafe's, theaters, clubs, and theaters and easy walk to the B Line to get downtown. But I suspect your profession will keep you very busy and you might not want to commute this far to Boyle Heights.
The area around USC seems like there are a lot of nice newer apartments. I like visiting the area from time to time because of the museums, rose garden, but I don't know about the safety of the area, and there are a lot of USC students. I also know it can get very congested because of all of the different entertainment/sporting events. But there is a nice little shopping village that opened a few years ago across from USC. It has a Trader Joe's, Target(City) and a number of places to dine and hang out. It's also close to the E line with easy access to Downtown and Santa Monica.
USC Village
Glendale, downtown and the surrounding neighborhood are walkable, and the area is quite bustling, but there is no train service other than the Metrolink so you would have to take the bus, but it's not really far from downtown LA. Again, traffic can be bad coming and going to downtown LA in the evening.
Last but not least, don't rule out downtown LA, especially the south park area. Nice new apartments, and you may find a decent studio. It seems. there are a lot more incentives right now to lease an apartment and I'm sure most of these units have washer/dryers and many would have plenty of amenities to entertain inside of the buildings even if you have a small studio apartment.
The San Fernando Valley along the Ventura corridor has some really nice and safe communities that are walkable. They are built for younger and single people with lots of apartments, but I think this would be too far for you to consider. Although if you ever decide to take a position in the valley, Sherman Oaks, Studio City would be my first choice and then Toluca Lake, Valley Village, on because these are further away from Ventura Blvd.