The damaged derelict in the home system on the prewarp start? I don't usually bother with it unless I have nothing else that I want my construction ships to be doing. It doesn't have a hyperdrive so it's not all that useful outside of the prewarp era, and it's usually far enough from the homeworld that it'll take a construction ship a long time to crawl across the system just to start working on it and then another long period of time for the construction ship and the derelict to crawl back to the homeworld if you try to do anything with it in the prewarp period. About the only good thing about repairing it is that there's a research bonus (of unspecified magnitude, unfortunately) for having construction ships repair technologically-advanced derelicts, and at the start of a prewarp game pretty much anything is advanced enough to give you that.
Other derelicts - both operational and damaged - can be worth keeping for fleet service, because they're usually quite powerful compared to the ships that you and your opponents can build at the stage of the game that you'd obtain them, but they're also worth disassembling at your shipyards at that point in time because there's a very good chance you'll get research towards some advanced tech (on lower research settings it can give you the tech outright; on higher research settings it'll 'merely' give you a decent chunk of research towards it). About the only thing that you "shouldn't" do with them is scrap them for money, because you get chump change for scrapped ships and they're worth a lot more as either powerful fleet units or as research boosts - though, being quite powerful and offering a chance of developing advanced tech, they also can also swing early-game difficulty quite a bit, and if you don't want that then scrapping them might make sense. About the only reason to blast them apart with your warships rather than repairing them is to keep them out of the hands of your opponents, though if you're dead set on not having anything that might swing the early game difficulty then blowing them up might make a degree of sense.