Its’ not a must to speak Spanish to apply for an internship in Spain, but there are some companies which require having a good level of Spanish, however there are some of them, especially in the area of international business or tourism which don’t require having any knowledge of Spanish.
Whether you know a bit of Spanish or you don’t speak a word, I highly recommend you to put energy to learn it.  Most of the people in Spain only speak Spanish and just a few speak very well English. Only at touristic attractions or international companies oriented and, in those surroundings, people know English but sometimes their English is limited.