My cousin's condo is in her mother's name. Her mother passed away. Now her father is now listed on the deed but not the mortgage. My cousin has lost her job and is moving in with her father at his condo. Can she just walk away from this condo in South Florida without it affecting her father's credit?

Abdicating her responsibility outright will not end well.

Typically, the parties that sign the loan promissory note are the parties that are obligated to re-pay the debt. By signing the mortgage or being on the deed, this does not obligate you to re-payment of the loan. However, in a foreclosure process the mortgagor will file with the local court to remove all vested owners from chain of title. This should not impact her credit if she was not a signer on the promissory note with the mortgage company.
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I am not an attorney, I only work in the mortgage industry. My statements are intended as legal representation and should not be taken as such. You should seek professional legal counsel for specific legal questions.

Let me get this straight. Your cousin got a mortgage on a condo that was in her mother's name? How did she manage that?

A lender is not going to loan money on property that is owned by someone else. That's why a mortgage is a secured loan instrument, it is secured by the property.

If dad inherited the property, he also inherited any obligations on it. Therefore, if the property is foreclosed his credit, will, most likely be affected.