In a way, this essay prompt is a fairly easy essay prompt. The reason for that is due to the fact that the response to the prompt is likely a strict "either or" response. You are either very much in favor of euthanasia being legalized, or you are not. This is a good thing in terms of thesis statement writing.
A thesis statement makes an argument that guides the following paragraphs, which try to support or prove it. With such a polarizing topic, making an argumentative statement should be relatively easy. For example, you could go with the following thesis statement. "Although there are reasons to legalize euthanasia, those reasons do not outweigh the reasons to keep it illegal to euthanize a human being." If you feel the opposite, then the thesis statement is easy enough to reverse.
The advantage to that kind of thesis statement setup is that it forces the paper to discuss both sides of the issue. This is going to be key in a topic that is as polarizing as legalizing euthanasia. You are going to have readers that are against your argument no matter what argument you make. It is important for you to admit the opposing arguments in order to get opposed readers to realize that you hear and understand their point. This will go a long way in helping that particular reader be open to your main argument.