Yes. Given the range of skills, James “Logan” Howlett (codenamed, Wolverine) has learned over his long life-span, swimming is something he would have learned how to do and used during his time as a member of the military during World War I, World War II and certainly as a covert operative for the CIA and eventually the Weapon X program. Under most of his military training protocols it would have been a requirement.
Let’s forget all of the pictures of him we see swimming, like when Jean Grey was believed killed and he was forced to tread water until they could be rescued. Or any of the other times we know he has had to swim for combat operations when he was first transformed into Weapon X. Wolverine can swim, no matter what uninformed writers and editors at Marvel comics may choose to believe. Does he like to? Especially since being given an additional one hundred pounds of adamantium to his physical makeup? Probably not. It certainly couldn’t make it any easier, even if it’s perfectly balanced and added directly to his bone structure.
It’s not a mattter of training or ability
Of course, Wolverine can swim. His body weight, while an impressive 325 pounds, should be no match for his superhuman levels of strength and stamina. He is capable of easily lifting 800-1200 pounds and running at speeds of up to 35 miles per hour for hours on end. His body produces limited fatigue toxins and those are processed immediately allowing him the ability to swim nearly indefinitely. 
The operative word is nearly. Eventually, if he were trapped out at sea, even he would drown through exhaustion and depending on what ocean and time of year, possibly hypothermia.
His body, even enhanced with its extra 105 pounds of adamantium, would be a perfect enhancement because it is at the core level of his being. He would have to work a bit harder, swim a bit stronger, and his core buoyancy would be affected but he would be using his raw muscular strength to overcome that. We have human SEAL team members who have been trained to swim with a hundred pounds of unbalanced and cumbersome equipment. His weight is completely internalized and already balanced. Wolverine’s problem with water is a psychological one.
Warrior, Berserker, Engine of Destruction
Logan is the living embodiment of combat. His body is filled with a genetic disposition for battle. Bones and tissues regenerate from nearly every injury, as fast as he can be harmed, within reason. A body with bone claws, superior physical senses, combat awareness sharpened by decades of training. Beserk rages allowing his body under duress to be even more powerful, more deadly, so even when he is overwhelmed by enemies he can KEEP FIGHTING! Swimming, and more importantly drowning, are directly opposed to his natural tendencies. Drowning is the antithesis of combat. You are alone, in the dark, with no one to direct your rage against. Oceans are large and almost impossible to cross under your own power. Now imagine being a man whose body is designed to fight against any threat and win, surrounded by the only enemy he can never defeat, no matter how undying he might be, this would be an enemy he would be psychologically unhappy to confront, no matter how little or much swimming he would have to do. If he has had to face the ocean more than once, his experiences could leave him simply unwilling to deal with swimming whenever possible.
Innate fear, magnified by his natural ability to survive
Wolverine hates the water because it can kill him, again and again, until he is rescued or until the trauma of drowning, one of the most traumatic ways to die, drives him insane. Drowning works against his nature, he is a being designed to struggle, and used to struggling effectively against any foe, struggling when drowning only increases the speed of drowning. From the way he talks whenever he talks about swimming, I am lead to believe he has experienced this trauma at least once. Remember, Logan will regenerate and return to life, even from drowning. The mechanics of that are frankly, mind boggling, but this might mean, he might recover, awaken, sink to the lowest point in the ocean, try to swim, drown again, reawaken, rinse and repeat; until he can reach the surface. Imagine that and then ask yourself, how close to the ocean would you EVER try to get knowing you might end up without a suitable flotation device?
Yes, the shoreline of a beach sounds just about right.
Imagine dealing with the horror of a shark based feeding frenzy and having the power to survive physical assault after assault, regenerating in salt water with limited food and the renewed threat of attack. Salt water is very painful to an injury. Doubt me? Next time you get a paper cut, dip that bad boy into some salty water and prepare for some salty language! Now imagine that over a portion of your body regenerating from a shark bite. Makes the ocean seem much less hospitable, doesn’t it. Yes, he will use his claws to kill sharks, but remember, if he draws their blood, they may still frenzy. A feeding frenzy isn’t likely to kill him but given his biological capabilities, it may make him very, very unhappy for a long time.
Yes, Wolverine can swim. He does so with trepidation because it is one of the only ways he can be killed, again and again and is completely dependent on someone else to save him. As the perfect warrior, his psyche has trouble processing this and if he has already survived a less than pleasant period awaiting rescue at sea, it makes sense for him to be at the very least, water-averse.
A Matter of Scale 