1960's well has groundwater entering
Extending the well casing above ground level is the standard approach, and works fine.

The only reason to put a (doghouse/windmill/wishing-well/lighthouse/etc.) over the extended well casing is for appearance sake (so you don't have to look at the bare well-head sticking out of the lawn.) The cap on the wellhead should do all the "keeping junk out of it" duties, the overstructure is just for appearance, if you choose. 

There's absolutely no reason to "abandon your well because it's old and drill a new one" here. The risks and expense of that will be far greater, to no purpose. I presume that you'll be having a well company do the work and they will have the proper parts and experience. If you are a gung-ho home excavator and weldor planning to do the job yourself, you do have a few more parts to buy than just a section of casing. Depending where you are, that may not be legal (my LAHJ requires a licensed well driller for actual work on the well itself.) 

Typically the wellhead was put in a "Pit" to keep it from freezing - you need a "pitless adapter" which goes through the side of the well casing and has a dovetail joint sealed with an o-ring that the pipe to the house connects to on the outside, and the pipe down the well connects to on the inside. That keeps the pipe below frost line. Power (assuming a submersible pump) is brought up to the cap. The cap seals the top of the well from entry of debris, and has a small screened vent (unless you happen to have a "real artesian well" where the cap needs to be a complete seal, and you don't need a pump.) It can be opened to access the well for maintenance, pump replacement, etc.