faculty position: personal statement in the cover letter of a job application

Yes. Search committees are very sensitive to what are called "two-body problems".

Failed searches waste everyone's time. Anything that suggests you may not accept an offer might be a warning sign for the department. This could be your spouse, because 1) people turn down jobs if their spouse doesn't want to live in that location (or can't find a job), and 2) two-academic households, they might only accept if offered two positions.

Your situation probably makes the department think you are more likely to accept. It doesn't sound likely to hurt your chances. However, there could also be some old-school sexism, etc., ("oh, your spouse works?") depending on the people and genders involved. I don't know the odds of this being a problem, but worth thinking about before disclosing.

Will this matter a lot? Probably depends on the type of school and how attractive the location is. If you are applying to a less prestigious university in a smaller city, departments might really appreciate the evidence that you want to be there (and won't move at the first chance). When I interviewed at a less-fancy school in my home state, I'm pretty sure that came up as a plus. If it's Harvard, I wouldn't expect it to matter at all.

In every department I've ever been in, a candidate having a spouse already employed somewhere within commuting distance (or any family connection to the area) would have been regarded as a big plus. Of course, "big" has to be viewed in context here, since obviously, it's quite small compared to the quality of someone's research and teaching. It will only make a difference if the department was seriously considering you already. But every department is worried about whether candidates will accept and retention after someone comes (Harvard included! the sort of people Harvard would hire have lots of other institutions falling all over themselves to make them offers), and the way humans make decisions is complicated.