Is there any writing on the black jacket?

I would peel back the foil on the end and take a close-up so we can see what is inside.

Can't tell what type of cable it is from the picture. COAX or small wire multiple conductors? Where is this in relation to the irrigation system components?

Hard to tell but looks like foil shielded coax. Probably cable TV.

Since I don't use cable TV, that might make sense. I'm thinking about just capping off the tubes to keep moisture out of them. What do you think?

It's probably an abandoned tv cable that was encased in the black tube to protect it :-)

I doubt that taking the time to cap them accomplishes anything. If it's ever used again, the cable company will simply run new cable or make a splice repair where you whacked it. Alternatively, you could just take it all out.