What is an appropriate term for being habitually put down by criticism?

Solution 1:

This is a condition called learned helplessness which began as psychological jargon but has become a subject of study in education and sociology.

Solution 2:

Believing what others say without ample criticism is called being naive. Naiveté is not characterized by insecurity, however: it's more of a failure to pay attention to or assign importance to crucial affairs.

Believing what others say because you are insecure is called just that: insecure. Insecurity is characterized by consuming self-criticism and self-doubt and indiscriminate acceptance of others' criticism.