The most current information is directly from DSB, Liste med trafikændringer, in particular the international page.
Today, Friday 11 September 2015, you can cross the border. DSB notes in a confusing paragraph:
International trains via Rødby F - Friday
The EuroCity trains are only in service between Copenhagen Central Station and Rødby F. Between Puttgarden and Hamburg the trains are cancelled. Please note that there are fewer trains in service between Puttgarden and Hamburg than normally, and the travel time to Hamburg will be extended.
They mean that the direct trains Copenhagen-Hamburg are cancelled between Puttgarden (the German border station) and Hamburg, but DB regional trains still operate (which explains why there are fewer trains).
In the Danish text, there is the crucial addition that it is now possible to walk from Rødby F to the ferry, from the ferry to Puttgarden, and take a regional German train from there. In the past two days this was not possible. For some reason this information is missing from the English text.
This method takes more time than a regular direct train. For your reference, the regular timetables for København-Rødby and Puttgarden-Hamburg are both online as PDF. There are regular regional trains from København to Rødby (which should be unaffected) 13:43–15:35, and 17:43–19:35, and an ICE 15:43–17:35 which may either run to Rødby or not at all. On the German side, there are regular regional trains (should run normally) 17:10–19:34, 19:15–21:50 (transfer in Lubeck), 21:15–23:50 (same). Other trains there are EC/ICE and probably cancelled. For the record, the last train from Hamburg to Berlin appears to be at 22:43.
In case you have time to travelling overland via Padborg, it is even easier:
International trains via Padborg - Friday
There are no changes to the timetable on the trains between Aarhus H and Padborg. There are only regional trains in service from Flensburg to Hamburg, which means the travel time towards Hamburg will be extended.
InterCity trains between Flensburg and Copenhagen/Østerport - Friday
There are no changes to the timetable on the InterCity trains for Flensburg. From Flensburg there are regional trains in service for Hamburg.
Note that if the train is cancelled, you can get a refund even if your ticket normally does not allow this. Also, considering recent circumstances, it is very likely that train-controllers will be lenient if you take a different train than you are booked on.
Finally, some trains are replaced by buses, but this is due to track maintenance and unrelated to the refugee crisis.