There are a number of waymarked trails criss-crossing Ladhope Moor. They're used by dog walkers, mountain bikers and horse riders, but you're allowed access even if you don't have any of these accessories...;-)
The paths form part of the network of waymarked trails around Galashiels. Not all of them are shown on OS maps, but many are shown on Open Street Mapping. There are also excellent booklets with maps available at tourist offices and they're also freely downloadable from Scottish Borders Council's website as PDF files. A series of them covers the entire region - highly recommended.
Please BYOP.
congrats to cloventrout - FTF on a wild & windy day...;-)
UPDATE July 2023: it seems that there are forestry operations being carried out. The area is not closed off but you should be sure to follow any restrictions/warnings that are put in place. As a general rule, weekends seem to be free of such measures. I'll check from time to time.