The writer of this sentence is making an unstated assumption (that readers are expected to understand) and is using "would" to indicate that this assumption is a hypothetical idea.
If I were to rewrite this sentence with the assumption "added", it would look something like this (the assumption is in blue):
Money should be the last thing on a personís mind and a public option for healthcare would solve this if money were on a person's mind (and if that were considered to be a problem).
Now to be honest, this writer's argument is a little unclear because the connection between money and public healthcare is unclear, and because the word "solve" suggests a problem but having money on the mind is not obviously a problem. So, let me give another example that should clarify this usage of would:
Anyone would be happy to win the lottery.
Again this sentence has an unstated hypothetical idea. If this were stated (again in blue), the sentence would look like this:
Anyone would be happy to win the lottery if he/she/they were to win.
I hope this is clear.