While easier, more accessible contraception would help, it wouldn't curb the reproduction of the most irresponsible people - those who don't think this stuff through at all, religious fanatics who believe God demands that they have more children whether or not they can support them etc.
In the short term such groups are pretty minor in Western societies in the long term we are all dead.
It depends on which kind of religious fanatics we are talking about. Mormons, Ultra orthodox Jews and Amish on average seem to take good material care of their children, setting them in a middle class trajectory with few if any starving and many of them assimilating as productive secular members into wider society. As long as society as a whole reaches a favourable equilibrium with enough of them assimilating this might not be a long term problem either.
I recently read an article by Steve Sailer that reminded me about something I have been puzzled by for a long time. Relevant paragraphs:
Poor people having fewer children means that the children have more resources available per capita making the children better off. Rich people having more children actually increases equality in society since it reduces the per capita resource advantage their children have. Rich people giving to their children is also one of the few cases where the redistribution of wealth doesn't reduce incentives for wealth creation. Rich people care about their children too.
Since programs aimed at reducing teen pregnancy rates do seem to have had some effect, we known something like this is possible without being horrible to the potential parents it targets.
Yet a policy of "poor people should have fewer children, rich people more" sounds heartless despite increasing general welfare both by making poor children better off and by reducing the privilege of rich children thus increasing equality which we seem to think is ceteris paribus a good thing.
Why is that?
Edit: To test the source of the reader's intuiton (assuming he shares it with me), I encourage the consideration of two interesting scenarios that may depart from reality.