As previously suggested I wash my hands immediately after handling bait or any fish, salt water is fine. 

You need to build a habit of rinsing off juices and bait after each hook is baited or fish cleaned.

Now days I rarely have a problem.

 

I believe it was bought about by years of baiting lines and wiping hands on a rag rather than actually rinsing them off.

This one....If the tips of your fingers are ok....you could get those jigging gloves which have the tips missing......that would protect your hands a bit ...yet allow your fingers tips freedom for tying knots....dont use squid for bait.........what fish are you targeting?.........i fished for 22 years in nz.......hardly ever used squid....mainly pillies and skipjack...KY.....stay away from whats causing the problem........Drink plenty of water(dehydration is your enemy and skin is where water is taken).....Heavy drinkers and smokers quite often have awful skin because the body sucks water from the skin(biggest organ in the body) to flush the poison out of the body.....that is why you piss alot when you drink booze !......the body sucks water from the skin as well as elsewhere to flush it out..........skin needs water and skin gets dry in the winter time.......hot showers and hot water to wash hands dries the skin.....i get dermatight arse in the winter when i have hot showers.....it wrecks the skin.........take a zinc supplement/fish oil to encourage faster healing...Stay away from alcohol until your hands are better......eat plenty of vit c.......once skin has healed ....use neutrogena hand cream after washing your hands or shower.......stay away from squid and your mrs will want your hands bro hahahaha.

Those fisherman wipes are also very good and contain tea tree oil which also heals the skin....i used those wipes all the time on the rocks....just kept one in my pocket and wiped my hands with it each time i baited up.......then at the end of day used one fresh one before i put my hands on the steering wheel.........only thing i ever found that could combat bonito/skipjack stink...........avoid the bottom one as well..not good for anything !..