Baker was admitted to Westminster School, on the foundation, in 1691, and thence elected to Trinity College in 1695 (B.A.
Baker rendered the master of Trinity great service by obtaining signatures in favour of the compromise between Bentley and Serjeant Miller in 1719.
His subservience to Bentley is ridiculed in The Trinity College Triumph: But Baker alone to the lodge was admitted.
Baker died on 30 October 1745, in Neville's Court in Trinity College, where, owing to financial misfortunes, he had ceased to be vice-master, and was buried at All Saints Church, Cambridge, according to directions given by him a few days before his death.
"He had been a great beau", says William Cole, the Cambridge antiquary, "but latterly was as much the reverse of it, wearing four or five nightcaps under his wig and square cap, and a black cloak over his cloath gown and cassock, under which were various waistcoats, in the hottest weather".