General encouragement:
* Boost: to increase or improve something
* Encourage: to give someone support or confidence
* Empower: to give someone the power or confidence to do something
* Inspire: to fill someone with the urge or ability to do or create something
* Motivate: to make someone want to do something
More specific encouragement:
* Affirm: to state positively that something is true or that you support it
* Champion: to support or defend someone or something
* Foster: to help something develop or grow
* Nurture: to care for and help something grow and develop
* Uplift: to raise someone's spirits
Words that suggest building up someone's confidence:
* Bolster: to make something stronger or more secure
* Strengthen: to make something stronger
* Uphold: to support or defend something
The best word to use will depend on the specific context.