How much does the choice of professors on my thesis committee matter?

For academic positions, what is relevant for your application is your letter of recommendations. Who is on your committee does not matter (except the head). Your job applications will not even list the members of your committee---this will only be seen by someone who looks at your thesis.

In academia, it's common that non-tenured (sometimes even non-tenure-track) faculty are on PhD committees. This is not viewed in any negative light. You want the most relevant people on your committee, so they can give you the best feedback, and will have a better sense of what you are doing to be able to write you recommendation letters. Often your supervisor can suggest who to put on your committee.

I imagine that non-academic employers will care even less, as they are (typically) not interested in your research.

The exact composition of your committee will probably weigh much less than the results you get (and any publications out of your thesis). Help in doing the work is more important than some final ceremony.