In general, small boats, i.e. higgins boats, backed off from the beach and returned to the ships they launched from in order to bring in subsequent assault waves. Landing craft were a valuable asset for logistical support in the days after the landing and were preserved when possible.

There are exceptions to this, of course. Some craft were stuck on the beach or holed by enemy fire and not seaworthy enough to return to the transport ships. I don't have any detailed information on the behavior of these crews.

There is at least one notable case of gunfire from larger landing craft being important:

In breaking this deadlock during the next hour, naval intervention played an important part. At about 1030 two landing craft, LCT 30 and LCI(L) 44, steamed full ahead through the obstacles off the Colleville beaches, firing all weapons at enemy strong points guarding the Colleville draw. The craft continued to fire after beaching. Not only did their action prove that the obstacles could be breached by ramming, but their fire, though failing to neutralize German positions in the Colleville draw, had at least a heartening effect on the assault troops.

(from US Army Historical Series: Cross-Channel Attack)

An LCI would have 4x20mm cannon or similar and an LCT 2x20mm cannon, so their fire would have been slightly heavier than machine guns. Close in support from destroyers 5 inch guns was probably more effective than landing craft fire, since 5 inch guns could actually destroy some German fortifications, not just suppress them.

At any rate, while circumstances varied widely, naval personnel were supposed to stay with their ship and either fight or keep bringing in troops and supplies, not go ashore and try to perform as infantry when they were not equipped or trained to do so.

Coast Guard personnel in many instances served as coxwains and crew members. I/Yaksarecool asked about this and to answer: Coast Guard personnnel served in all theaters of the war, and perhaps the most recognizable role they played was as boat crews, considering their expertise in maneuvering in close aboard and shallow draft, they were ideally suited for the role they played.

As a member of the Coast Guard, I can tell you that this is a tradition which the service is very proud of and chooses to educate members on while they are processed through boot camp. Douglas Munro, the only Coast Guard recipient of the Medal of Honor, was killed in action in the Pacific (Guadalcanal, if I remember correctly) while providing covering fire and positioning his craft between Marines that were evacuating the beach and the enemy firing positions that were targeting them. Petty Officer Munro was a Signalman, and his job was to direct landing craft to the beach.

The Coast Guard combat contributions tend to be forgotten by the general public and other branches, but in WW2, I believe they really made a name for themselves.

Additionally, there was a quote from a Coast Guard veteran of Normandy which I saw in this thread that mentioned the use of Springfield rifles. This is not surprising to me in the least. Marines like to complain that their equipment is the hand me down of the Navy, and while that may be true, it is certainly doubly so for the Coast Guard. On my previous assignment our armory had M-16A2s, while the Army and Marines are on the A3 and A4 variants. Some things seem to never change.

I personally wondered why the boats were designed to open in the front?

If they instead opened on the sides, the enemy machines guns would be only to hit one side or the other.