Best place for dog owner to live with walkable commute to Montg St BART?

San Francisco is a very dog-friendly city! You shouldn't have any issues finding an apartment that will take you and your dog, but the price constraint is going to be the bigger issue. I like the South Park recommendation, but the apartments are typically expensive. Living as far out as Glen Park (they have a BART station) is going to be much more cost effective (you won't have an issue with your budget), and the whole neighborhood is filled with parks and dogs!

I think you would really like the South Park area between 2nd, 3rd, Bryant and Brannan. There's a great park there as well as excellent bars, restaurants and shops.