If my elderly mother sells her home and uses the money to rent an independent living apartment, what is the best way to handle the money?

What is the down payment and monthy charge for a large one bedroom apartment?

You may want to consider a living trust. This protects her assets to some degree. (there are great articles on the subject on this site.)
Currently, there is a "look-back" period of 5 years, should your mother ever need Medicaid assistance.
If you are asking about "management." I have created accounts for Mom at our local bank. I set up an online bill pay account so I can manage her funds and pay bills...such a timesaver.
Also, do you have your Mom's PoA for both medical and financial? You may also want to become a "signer" on her bank and credit card accounts in case you need to pay a bill or purchase something for her. Do not place your name on any account or investment as 'joint' unless you are prepared to take financial responsibility.
It is always a good idea to consult an elder accountant or lawyer for more complex issues.
good luck