Where is a good place in Los Angeles to rent a home for less than
$2,250 that is an easy commute (less than 30 minutes) to Pasadena, has 
good air quality (relative to Pasadena) yet is fairly close to the beaches 
for weekends. Home should be 3+ bedrooms, have excellent schools and
walkability to safe parks and/or nature trails, and be located in a
clean and politically liberal neighborhood.

daedalus1-ga, the closest communities to Pasadena, on the L.A. side,
is Highland Park (affordable) and Eagle Rock (more expensive). 
However, both are about 15 miles from the beaches (as the crow flies).
 Assuming that Pasadena is where you will work, and the beach aspect
is recreational, I recommend concentrating on the work aspect because
commuting is a major concern (time-wise).

Why would you want to live in a "politically liberal neighborhood"?

I recommend the eastern part of Studio City - near Universal studios.
It's 20 minutes from Pasadena and 30 minutes from the beaches with
good schools. I emphasize the eastern part so you'll have access to
the 101, 134, 5, 210 and 405 freeways within a 5 minute
drive...there's actually a house available for rent right now within a
small community called the Island..it's across from Universal Studios,
from Lankershim turn on Valleyheart. Very few people are aware of this
small community. Another area is Toluca Lake but not as familiaar with
the schoools...If you're interested in specific details about this
rental post a reply and I'll get you the details.