Helping someone climb

Per Wiktionary, the word you are searching for is to give so. a leg up.

Leg up: the act of assisting another's progress over a wall or other obstacle by forming a step for one of their feet with one's hands

To boost or give someone a boost may work well, depending on how formal the context.

At least in American English, giving someone a leg up often is used in a more metaphorical sense, as in giving someone an advantage of some kind which is not necessary physical, so this phrase may confuse some readers if you are actually referring to the physical act of lifting someone up.