Cook, 31, said he has "questions" over his role after England went 3-0 down after the fourth Test to lose the five-match series against India.
Despite suffering five defeats in the last seven Test matches, Stokes told BBC Sport: "As a leader he's been brilliant and he still will be.
"He doesn't want anything except success for the team."
Stokes, who has played all of his 31 Tests under Cook, added: "Over the last two years, he's been absolutely fantastic.
"He has been part of the driving force of getting us to be the team we are today. Albeit we're 3-0 down, we've made huge strides since Australia (in 2013-14).
"Whatever happens, I'll have a lot of respect for him and whatever decision he makes - but I hope he sticks with what he's doing, because I think he's doing a great job."
Cook, who turns 32 on Christmas Day, took over as captain from Andrew Strauss in 2012 and has led England in a record 58 Tests.
He presided over the Ashes series wins of 2013 and 2015, but came under intense pressure after the whitewash in Australia in 2013-14 and the following summer when England suffered defeats by Sri Lanka at Headingley and India at Lord's.
After the latest loss, Cook said batsman Joe Root is "ready" to take over as captain, but the left-handed opener will discuss his role with now director of cricket Strauss.
Coach Trevor Bayliss said the speculation was a "pain in the backside".
All-rounder Ben Stokes hopes Alastair Cook stays on as England captain because he is doing "a great job".