Do you know what type of hydrangea it is? This is not a good time to prune some of them. If unsure, wait until early to mid May and first cut off at the base/bottom any stems that appear dead and have no leaf out. Be careful with stems that leafed out; flower buds usually grow at the end of the stems so pruning stems that leafed out will cut off this year's blooms.

Next, you can rejuvenate the hydrangea by cutting the stems at the base/bottom in three phases. On year 1, cut off the oldest stems. On year 2, cut off the next 1/3 oldest stems. On year 3, cut off the rest. If you cannot tell the oldest from others, prune a 1/3 section of the bush this year, another 1/3 chunk next year and the rest in year 3.

This can be done after the plant has bloomed but before the end of June, so you get to enjoy some bloomage this year. If you do not care about the blooms, you can start to prune in thirds now.

Luis

By the type of leaf and the growth habit, these are macrophyllas or big leaf hydrangeas. Some macs will bloom on both new growth and old growth but the vast majority bloom only on previous years' growth. That's why Luis' comments about the timing of pruning is so important.

Just to expand a bit on his comments - the rejuvenation pruning (or the 1/3 removal of oldest growth annually) is the best way to reduce the size of the shrub while still maintaining the flowering potential. At best you'll only be removing 1/3 of the flowers :-) But you can engage in a much more radical approach by cutting the plant back hard. This virtually will eliminate any flowering for the season but you will immediately get a significantly smaller shrub.

And the 'end of June' caution is highly dependent on where you are located. In my area, hydrangeas don't even start blooming until July but continue producing flowers until well into fall (actually until frost, sometimes beyond) so waiting to prune them after the bloom cycle means you'd be pruning very late in the season as well as cutting off next season's flower producing growth.