Why would you want a smaller critical region when testing large samples? Is it because when you have a smaller critical region you have a smaller probability of the event happening? 
 A: There's a tradeoff between the two error types (type I and type II) -- at a fixed sample size (and all other things being the same) -- if you make one error rate lower the other will be higher.
Imagine you have some particular effect-size you'd like to be able to pick up with reasonable probability (a specific one you've got in mind), at some type I error rate; so you take some sample size $N=n_e$ that will achieve that desired power (equivalently, make your type II error below some threshold).
Now let us increase our sample size progressively. If we don't change our type I error rate, all the impact of the larger sample size goes into reducing type II error. So it goes down and down. At some point you might be looking at the next increase in sample size taking the type II error rate at the particular effect size from say 0.001 to 0.0005 ... but since your type II error rate is already miniscule, you could instead use some or all of the impact of that bigger sample in reducing type I error instead.

Now if you don't think low type I error is important, why not make it larger in the first place and reap higher power (lower type II error)?
On the other hand if both types of error matter to you there must be a relative cost to making those kinds of errors -- and whatever tradeoff between them you choose, those relative costs don't change as you change your sample size, so presumably if you chose them at about the right relative sizes at the initial sample size, you should be seeking to keep them roughly in proportion as sample size increases.
