What is the duty of the boatswain in The Tempest? Is he doing his job well?

A boatswain's duties on a sailing ship haven't changed much since Shakespeare wrote The Tempest.
The boatswain is the senior crewman on the deck whose responsibilities include managing the deck crew, maintaining the deck and the rigging of the ship (lines and sails), and managing the anchor.
One of the boatswain's duties is to keep the ship afloat in a storm, something that the Boatswain in The Tempest apparently failed to do. There were, however, extenuating circumstances, real and imagined, which caused the Boatswain to appear to fail in his duties.
First of all, the reality of the situation is that there was no storm. The "tempest" was an illusion, an elaborate magic trick, devised by Prospero and carried out by Ariel.

PROSPERO: Hast thou, spirit,Performed to point the tempest that I bade thee?
ARIEL: To every article. (1.2.228–230)

The crew and passengers were never in any danger.

PROSPERO. But are they, Ariel, safe?
ARIEL. Not a hair perished;On their sustaining garments not a blemish,But fresher than before . . . (1.2.340–342)

The ship, which appeared to be split in half by the wind and waves, was not in any way damaged by the storm.

PROSPERO. Of the king's ship . . .
ARIEL. Safely in harbourIs the King's ship, in the deep nook where onceThou calld'st me up at midnight to fetch dewFrom the still vexed Bermudas, there she's hid; . . . (1.2.264–270)

The Boatswain was happy and probably very relieved to report in act 5 that the ship was in as good shape as ever.

BOATSWAIN: . . . our ship,Which but three glasses since we gave out split,Is tight and yare and bravely rigged as whenWe first put out to sea (5.1.258–261)

In the face of the imagined circumstances, the Boatswain did remarkably well, With the "tempest" raging around him, the Boatswain remained remarkably calm, and he ably managed the crew, even with the constant distraction and interruptions of the King of Naples, Antonio, Sebastian, and Gonzalo, who seriously interfered with the Boatswain and the ship's crew doing their jobs.
The Boatswain did everything he could to save the ship, under the real and imaginary circumstances, for which he should be commended, particularly considering that he was going to lose the ship no matter what he did, because Prospero had already planned it that way.