If you can believe the claims, cuts covered with a band-aid heal faster than cuts left uncovered. Similar claim for Neosporin - cuts heal faster when this is used.
I believe the band-aid claim as I have observed the difference myself in a decidedly non-scientific, uncontrolled study. I don't use antiseptics so can't speak to that claim.
My experience is exactly the opposite. My cuts heal more quickly dry and exposed to air, once an initial bleeding, clotting and scab forming has taken place.
I've also observed that cuts heal more quickly when they are allowed to dry and form a scab. Application of a salve keeps the wound moist and scab formation is delayed. However there are good reasons to apply an antiseptic or antibiotic salve, when infection is a concern.
I remember a Dr. Oz show where he recommended triple antibiotic cream and a bandaid, the key being keeping the owie moist and not forming a scab. I admit I usually go with letting a wound form a scab.
Oddly, I have little faith in the advice of Dr. Oz.
To be clear, my comments refer to minor wounds, what we sometimes call "owies." More severe wounds require different treatment than superficial wounds.
For example, my second cut has almost healed, near 99%. It was the less consequential of the two.
My more serious cut (treated with hydrogen peroxide) bled out under the Band-Aid and kept itself fairly moist. Today I scrubbed off the excess blood (on the skin around the wound) and now I observe an exposed clot that has not formed a scab. I'm drying it out in air right now but will apply a Band-Aid before bedtime to keep the wound clean, and to keep debris out of the wound.
I think the best practice for superficial wounds is to clean them off with an antiseptic and then apply a Band-Aid. Once a scab has formed I would just leave it exposed and allow it to heal. If I feel there is substantial potential for accidentally re-injuring the scab then I would protect it with a Band-Aid.
Again, nothing I'm referring to has anything to do with anything other than superficial, small wounds.
I think isopropanol (IPA) is the way to go for these small injuries.