Currently developed countries are all struggling with fertility rates below the replacement rate, so while something like curing aging would certainly lead to those sorts of problems, I would expect extending healthy lifespan by some decades could actually solve more problems than it creates. Also as health and lifespans increase, people tend to delay having children, and have fewer overall.
Also thinking a bit bigger picture, even if people didn't die of old age at all, couples could still have approximately 1.1 children each on average without the population increasing beyond ~1.5x its initial level. Of course, even that level of restriction seems far-fetched (but then, so does a cure to aging), but it's not like people would need to stop having children entirely.
Can't edit: the 1.5x was wrong. Assuming no one dies, if every couple has 1 child, population would approach a limit of double its initial level. Since some will die in accidents, you could have slightly more than 1 child per couple without unbounded population growth.
Also thinking a bit bigger picture, even if people didn't die of old age at all, couples could still have approximately 1.1 children each on average without the population increasing beyond ~1.5x its initial level. Of course, even that level of restriction seems far-fetched (but then, so does a cure to aging), but it's not like people would need to stop having children entirely.