Neither is very good as doing any conversion on a column will at best use an index scan. The best solution is to store data in a properly typed column.
There are other problems with both. With converting the date to a string you need to make sure that they are both in the same format and how can you guarantee that the string date column has a consistent format. When converting the string date column to a date you may get an "invalid date" error and the entire batch will fail because you can't guarantee that you will get a value that evaluates to an acceptable date.
All that being said, if I had to do it, I'd probably convert the date to string to avoid the batch failing. You won't be guaranteed that you will get the correct data returned, but you will get something.