Not an easy subject. I am guessing that most of the responders have not had a serious or very long term problem, because some of the proposed solutions are a bit anodyne. For those that have a more serious relationship with H, here are some things that may be worth trying. Warning: pretty, it ain't...
1. Cleanliness. Wash with fresh water after every crap. If out and about, use your cupped hand and fingers with tap water, or the flush water in the loo. I know it seems gross but it is clean as it enters the bowl. Dry gently, as stated elsewhere.
(It doesn't get better)
2. Push any prolapsed hemorhoids back in. Use saliva. The mucosal part of saliva comprises glycoproteins, the perfect lubricant for your middle-finger.
3. Ensure you leave two hours after a crap before going for a run. This allows whatever is inside the anus to revert to its normal mode for keeping things in - so there is less chance of anything prolapsing.
4. Avoid all greasy, oily preparations (including specialist H products) internally, especially before exercise. If you have got this far, I am assuming you really do have hemorrhoids and that the internal ones can occasionally and inconveniently poke out. Lubricants exacerbate this tendency.
5. You can use limited amounts of H products and moisturising emulsions externally if you have external hemorrhoids, since they can prevent irritation.
6. Do not use pain relief creams intended for topical application for muscle pain. Most will burn...
7. Ice is great for relieving and reducing inflammation. Hold an ice cube in place, e.g. after showering. When small enough, you may want to insert it. (You were warned about this being for hardcore H sufferers). Also good is a small ice-gel pack. Wrap it in a piece of kitchen role and get it close and comfortable for ten or fifteen minutes.
8. And diet? Usual advice. Lots of fibre and starch. Go easy on alcoholic excess and rich foods, or be prepared for flare-ups after indulgences.
'Too much detail'? Apologies; but these have worked on a long term basis to avoid medical interventions, and I am still running.