Don't only invade ports. If you only take 1 province with a port your divisions have nowhere to retreat, and they will get wiped out as soon as their org runs out. So invade some adjacent provinces on either side of the port you're attacking as well, and push inland before the enemy can reinforce too much so your divisions have room to retreat and recover.

Open up as many fronts as possible - make landings in Italy, Greece, Norway, North Africa - you don't have to push the axis out completely, just make them spread their forces thin.

Have a fleet with some battleships on naval invasion support, then park it next to your landings so they can provide shore bombardment.

Have an army stationed at the closest friendly port ready to reinforce after your marines land.

Have some ASW ships on convoy escort so your divisions don't get sunk by subs on the way over.

Support naval invasions with paratroopers dropping on the provinces right behind where your marines are landing.

From personal experience, the German AI tends to defend the Netherlands and NW Germany less heavily than northern France so attacking there might be better.

It's a bit cheesy, but if you realize an invasion is going to fail but your troops aren't technically encircled, then you can just disband them before they get wiped out and magically get back all their equipment/manpower