Database index on FK with only NULL values
The only situation where an index with full key cardinality = 1 will help is when you are looking for something that does not exists. By using such index you can quickly conclude that nothing will match your predicate, whereas without the index you will have to scan the table.

If you have a large table and you frequently look for things that does not exists, you can benefit from such index (fairly rare situation).

The downside of such index is that the only leaf-page will have a long rid list that has to be maintained during modification of the table. 

In most cases you will see an overall performance improvement by dropping that index, but it is impossible to tell for sure without having access to the system. Ideally you have a test system where you can run a representative load for your system and compare the performance with and without the index. 

For the big picture, what is the purpose of the attribute that never exists? You may want to consider normalizing this table and move this attribute to a separate table.