KASTRUP, Denmark—Sweden began enforcing tighter border controls Monday to curb the influx of asylum seekers, prompting Denmark to begin similar checks in a further weakening of Europe’s principle of open borders.

Sweden warned weeks ago that it would impose systematic identification checks at its borders, saying that the country of close to 10 million people was already straining to cope with the estimated 160,000 migrants who arrived in the Nordic nation last year.

...