If I claim my mother as a dependent on taxes will it effect her Social Security or Medicaid/Medicare?

It won't affect her Medicare or Social Security. For Medicaid, that would depend on your state and what they allow. If you provide services for her that Medicaid would likely have to pay someone else to do, they may allow that. Be prepared to "make your case". You may want to check with a CPA on this before checking with your state on Medicaid. Good luck. I hope you can. Family caregivers provide many services for our vulnerable loved ones that save the government billions of dollars.
Take care,
Carol