The classic barbell bench press is an internal rotator of the shoulders. Due to the constant seated/hunched position many people are in for the most of the day, the shoulders become excessively internally rotated. Unfortunately, the bench press promotes further internal rotation of the shoulders.
Couple that with the fact that the barbell doesn't offer much freedom of movement in the glenohumeral joint, it's no wonder so many people complain about shoulder pain.
1. Swiss Bar Floor Press
Any floor press variation you do limits the range of motion your shoulders have to go through, which is beneficial from an injury and pain-prevention standpoint.
You can use a neutral grip with the Swiss bar, which externally rotates the shoulders (as opposed to the internal rotation you'd be using with the straight bar). You also have the ability to load more weight than you potentially could with dumbbells.
2. Dumbbell Floor Press (with External Rotation)
You'll use a reduced range of motion in the shoulders and engage your triceps more with this variation. Turning your palms in toward each other externally rotates the shoulders and alleviates pain/unwanted tension in the shoulders as you press.
3. Dumbbell Press (with Slight Decline)
Slap a plate or two underneath the front of the bench so you're on a slight decline angle. This reduces shoulder stress and emphasizes pec engagement during the bench press.