The managers are not covered by a union. Only players on a team’s 40 man roster or 60 day disabled list are members of the union. Managers and coaches are salaried employees of the club. Each probably has a clause that states whether or not they get paid in case of a work stoppage.
Managers get paid during strikes. After all, they are management!
If an owner didn’t have the cash to pay the manager, he would likely be laid off just like in any other labor situation.
So if Steinbrenner starts running short of cash, he will have to decide between Joe Torre’s salary (which is not inconsiderable) or buying another ivory-handled back scratcher.