big, considerable, enormous, great, marked, significant, substantial, tremendous
growing, increasing
chief, dominant, major, overwhelming
powerful, profound, strong
He had a profound influence on modern poets.

subtle
important
crucial, decisive
potential
continuing, enduring, lasting
improper, undue (law)
The court found that the bank exerted undue influence over Mrs Black in getting her to sign the contract.

disproportionate
beneficial, positive
adverse, corrosive, corrupting, destructive, disruptive, evil, malign (formal), negative, pernicious
moderating, stabilizing, steadying
calming, restraining
civilizing
direct, indirect
causal
mutual
relative
This paper evaluates the relative influence of religions.

far-reaching, pervasive, wide
external, outside
The religious community wished to be independent of outside influence.

peer
the effects of peer influence on crime

parental
foreign
cultural, economic, genetic, political, social, sociocultural

have
gain
give sb
Her wealth gave her influence over affairs of state.

assert, exercise, exert, use, wield
Can you use your influence with the director to get me a part in the movie?
Drug cartels wielded enormous influence in the city.

retain
lose
expand, extend, increase, spread
The unions have been able to extend their influence over all industries.

minimize
eliminate, remove
limit
diminish, reduce
counter, counteract
avoid, escape, resist
be under
The court was told that he was under the influence of alcohol when he was stopped by the police.

come under, fall under
She came under the influence of Sartre at this period.

be independent of
demonstrate, reflect, reveal, show
Spanish architecture shows Moorish influence.

acknowledge, recognize, understand
analyse/analyze, assess, determine, evaluate, examine, explore, investigate, study
They examined the influence of farm size on technology adoption.

attribute sth to
Much of his writing can be attributed to the influence of Freud.

trace
It's easy to trace the influence she's exerted on contemporary artists.

extend
Their influence extended as far as China.

grow
The Internet's influence is growing every day.

wane
His political influence was waning.

under the influence
The town grew under the influence of colonialism.

influence from
There was no influence from outside.

influence in
She has a certain amount of influence in the way things are organized.

influence on, influence upon
They were a major influence upon the development of the sport.

influence over
I have absolutely no influence over him.

influence with
He used his influence with local officials to gain commercial advantages.
Queen Isabella was urged to use her influence with the French monarch.

a sphere of influence
Rome's sphere of influence extended as far as Scotland.

bring your influence to bear on sb/sth
She tried to bring her influence to bear on her husband.

under the influence, under the influence of alcohol, under the influence of drink (BrE)
He was arrested for driving under the influence (= drunk).

big, considerable, great, significant, tremendous
dominant, major, overwhelming
powerful, profound, strong
important
obvious
lasting
early
Who were your early influences?

formative
His uncle had been a formative influence from his earliest years.

diverse
He is a writer who draws upon diverse cultural influences.

outside
Parents often seek to shelter their children from outside influences

good
adverse, bad, corrupting, destructive, disruptive, harmful, malign (formal), negative, pernicious
moderating, stabilizing, steadying
calming, restraining
civilizing
environmental, genetic
artistic, cultural, intellectual, literary, musical

be
have
The band had many influences.

be exposed to
a study of children exposed to different cultural influences

draw from, draw on, draw upon

be at work
There were a number of influences at work in Wright's architecture.

affect sth, shape sth

influence on
She's by far the biggest influence on my writing.