To class up something means to make that thing have class, or to make it classy:
: having or showing class: such as
a : elegant, stylish • a classy clientele • a classy hotel
Actually, there's an entry on Urban Dictionary:
class up the joint
To add a touch of refinement or elegance to an otherwise seedy or unsavory location, grouping, or endeavor.
The crystal doorknobs in the dive bar really 'class up the joint'.
So he's saying that he doesn't know who makes the place more elegant, the lady in the fur stole, or "cookie". Cookie appears to be Carl's nickname, and I'm guessing it's derogatory.
By the way, it appears that he's being sarcastic. The speaker doesn't like the woman because I think she was inebriated at the time and later he says, "spoiled little rich girl gets her kicks
draggin' stray men in here", suggesting that she might be promiscuous. Judging from the script, the speaker doesn't like African-Americans either. This suggests that Carl isn't welcome, and so neither Carl nor the girl would actually class up the joint, in the opinion of the speaker.
The movie was set in the 1940s, when racism was much more overt and widespread. If calling Carl "colored" is the language used in the film, it makes sense because of the setting. It might have been an acceptable term back then (I'm not sure). However, nowadays, it's considered highly offensive. Instead, black or African-American are generally acceptable terms in the US.