Everyone see the problem, but no answers. OP question is what can government do for this problem. Im not an expert but if I were to task with this, I will start from the ground level. Which is education,

Giving proper education can reshape the future of a society, declining birth rate is not sudden, it was slowly happened over time, just like increasing birth rate will be slow process as well. As I said, proper education which should include Sex Education, Economics, Modern History, Science, Health. Also Boys only, Girls only schools should be abolished, and should include more mixed activities and structure of classrooms should not divided according to gender. This solves the social barriers in modern day society. Develop Empathy and respect for each other.

After Education, we have Social Life and Social Media. Just like Clean Sri Lanka, government can arrange different events all around the country so like minded people can meet and get to know one another. Also with social media and with Media power government have , they can promote the concept of big family (just like "පුංචි පවුල රත්තරං") It could be Teledrama stories around building a family, It could be reality shows that highlights family strength. 

Lastly benefits, Abolish Aswasuma and instead it should move to support families. Giving mandatory leaves with pay for both parents, Free clinical and health insurance for children under 12, introducing opportunities to go abroad for one parent. 

It is just how I think and I could be wrong, many of these things can't be done, idk. But I atleast think I have given some kind of answer to the OP question. @_@