Your numbers are factually incorrect.
In Denmark, the number of asylum applications in 2015 was 18,492. That's applicants, the number actually granted asylum is significantly lower. So far, it seems the trend for 2016 will be less Syrian refugees (as most of those who are able to flee have already done so), and more refugees from comparatively peaceful countries in Africa and Asia (e.g. Afghanistan). Where almost all Syrian asylum seekers received asylum, that is not the case for refugees from other countries. As such, while the number of applicants might rise, the number of people actually granted asylum in 2016 is not expected to grow significantly.
When you then factor in that most of the expenses of receiving and processing these refugees is paid from money already allocated to foreign aid, it becomes quite clear that no, asylum seekers are not making a significant dent in the overall budget.