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".

The decision comes after a report by the UN cultural agency Unesco said the area "should be off-limits to commercial logging in its entirety".
The Tasmanian Wilderness covers about a fifth of the island and is one of the world's last big temperate forests.
Conservation groups have welcomed the Unesco report and the logging decision.
In 2014 the Australian government asked the agency to revoke the special status of the forest to open up parts of it to logging in order to boost the island's economy, where unemployment is above the national average.
But the move - by the government of former PM Tony Abbott - attracted much opposition from environmental groups.
The Unesco report released on Saturday said the organisation "does not consider a World Heritage property recognised for its outstanding cultural and natural values the place to experiment with commercial logging of any kind".
The Tasmanian Wilderness is home to ancient forests, some of the tallest flowering plants in the world and is a stronghold for several animals that are either extinct or threatened on Australia's mainland.
Both the Australian federal and the Tasmanian state government have said they will abide by that recommendation.
All-rounder Ben Stokes hopes Alastair Cook stays on as England captain because he is doing "a great job".
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The Australian and Tasmanian authorities are abandoning their bid to have logging permitted in the Tasmanian Wilderness, a World Heritage site.