Perhaps you could spray cleaner (oil based or Pledge or Endust) on a dry cloth and use this to lightly go over the blinds. My Mom always used a dry cloth with spray. Her furniture looked good for 60 years using this method.

Forget vinegar. Dirt deposits have an oily component that vinegar cannot clean. Might as well use plain water, since vinegar smells bad. Here's what I do:

- Swiffer dusters if there's just loose dust. (By the way, those specialty blind cleaners that look like 5 fingers don't work well, IMO.)
- Mr Clean Magic Eraser for more serious grime, OR (if I don't have MCME on hand)
- microfiber cleaning cloths (I use the automotive ones from Costco, have dozens on hand) with Method All-purpose cleaner (really good performance, without the nasty fumes of 409). OR, if the dirt doesn't come off easily with Method, then Trewax Nature's Orange Degreaser, which is the BEST heavy-duty household cleaner I've ever used...however, I can't find Nature's Orange anywhere anymore (probably due to VOC laws) and products that look similar are probably different formulas and not as effective....so then I'm back to Mr Clean Magic Eraser.

I don't have tapes, mine have little strings. I would try scrubbing on Oxi Clean I had mixed in heated water (200*) with a microfiber cloth. Give it time to work. Then maybe a shot of steam from a steam cleaner. Try not to steam the wood, and definitely don't leave any moisture on the wood. Then, soak up as much moisture as possible from the tapes with dry microfiber. This is assuming the tapes are white. If they are colored, I would measure the Oxi carefully according to package directions, not heat it as aggressively, and test a small inconspicuous area first. Heated, concentrated Oxi Clean CAN remove color from dyed fabrics/rugs.

Once they are clean, regular dusting with Swiffer should keep them that way for a long time.