How can I repair or replace this rusty leaking bathroom sink handle?
You probably have a leak in the supply line or the valve stem itself.  

It is possible that the supply line is loose and water is leaking and pooling under the faucet.  If that is the case, tightening the supply line, or resetting the conector with new teflon tape  may solve it.  

It is also possible that the faucet is seriously damaged. If so, you will probably need to replace at least this faucet section. Becaue faucets are sold as sets. it may be more practical to replace the whole set. 

The first thing you need to do is turn off the supply valve sending water to this side of the faucet.

The faucet handle should have a cap a the very top that can be pried off with a thin blade, such as a butter knife or a utility knife.  Under the cap should be a screw that should be loosened that will allow the removal of the handle itself.

After that, you can get access to the valve stem itself. If you turn on the supply valve briefly, you may see water leaking from some part of the stem.  

This section of the faucet can usually be removed by loosening a nut under the sink, after the supply valve is turned off and the supply line is disconnected. You may need a special wrench called a basin wrench.

If you are unfamiar with changing faucets, you may want to consult a plumber or a more experienced friend.