How do I maintain my rose bush leaves that are getting brown spots?

It could have a fungus. Use a there in one systemic control by Bayer. It is a fungicide,insect prevention and fertilizer in one.

It sounds like it could be black spot disease. You can try to keep that in check first, by pruning away the infected leaves and infected stems. Also, rake up all the fallen leaves under the plant. Throw the infected debris away, rather than putting it on your compost pile. Then treat with a fungicide. I'm a fan of organic so I'd suggest trying Neem oil or copper.

Too much water

Could be too much sun or insufficient water, depending on location. Try misting?

I would try just one of these suggestions at a time and then you can know for sure what the problem was. Good luck!