does renting an apartment affect the process of me helping my husband get papers?
No, it does not.
Your question is unclear. Are you a US citizen? Are you petitioning for your husband to obtain a green card? If you are asking about whether or not renting an apartment in both of your names will affect that application, for example, the answer is yes and it is a good idea. Otherwise you should really provide more information or call an experienced immigration attorney. Feel free to contact me via email.