I feel that people overestimate and under estimate the effect forcing the darndelles and knocking the Turks out of the war would have. It definitely would have had a massive effect but its not a panacea that would solve Russia's issues.WRT trying to force the Dardanelles - I was under the impression there were whole lot of Russian merchant ships filled with goods that they could not export because they were stuck in the Black Sea. Is this true and would the economic benefits to Russia from forcing the Dardanelles be enough to alleviate the pressure that was on the regime to the point where the perhaps the downfall of Imperial Russia is avoided? I've heard this argument put forward but I do not know if there is serious analysis behind it or if it just sounds nice.
However, if this is true, then that campaign was worthwhile from a standpoint of strategy and it's a crying shame it did not work out.
Remember the Russian railway net was operating at full steam to support the war and that internal trade suffered massively. The biggest causes of both the February and October revolutions were food shortages In Moscow and St Petersburg due to a lack of transport capacity.
Due to this lack of transport capacity there's only so much that can be imported and so much that can be exported.
It would definitely improve the Russian position (both opening trade and releasing soldiers for other fronts). How much is a topic of debate and argument. My own estimate (and only an estimate) is that Imperial Russia still falls but does so six months later.