Did the Chernobyl accident affect the planet's environment?
No.
One of the major benefits of nuclear power is that the waste, although very dangerous, is contained relatively easily. As long as they can keep the spent fuel rods from, say, falling into the ocean or getting into ground water, there shouldn't even be any local problems.
Even if the power plant had a complete meltdown, the core of a nuclear power plant is surrounded by sand. The sand would melt, encasing the core in glass, and it would slowly cool over a long period of time.
So yeah. There most definitely won't be any global environmental effects.
PS: This is why we need to start using facilities like Yucca Mountain to contain spent nuclear fuel. If Japan had been storing their spent fuel rods someplace secure instead of just leaving them at the plant (kind of like what we do with our spent fuel now) this wouldn't have gotten as bad as it is now.
Then again, they do have to store the spent fuel on-site until it cools enough to transport, so maybe it would still have been a problem.