What sort of suicide are we talking about here? Euthanasia/physician assisted suicide or just suicide in general. The framework changes a bit.
I support euthanasia, especially in the case of terminally ill patients. I think a human being has the right to end their life with respect and dignity. Although, I think second doctor's opinions as well as counseling should be mandated before making the decision to ensure people don't act hastily.
Even if not terminally ill; I cannot imagine living in a vegetated or incapacitated state. I don't think society has the right of forcing one to live in indignity and embarrassment. A living will drafted by a reputable lawyer should be sufficient. Good lawyers usually advice their clients to speak to pastors or counselors to ensure clear conscience before drafting end of life treatments.
Now suicide by healthy, able bodied individuals who have a long life ahead of them is very unfortunate. Sadly bullying, bankruptcy, job loss, divorce, affairs, loneliness and various other stress factors lead to depression that leads to suicide. They do have the right to end their lives and they are not criminals. But these individuals are also not their normal selves and would not consider the option if they were in their normal state of mind. They do need help. I view criminalizing suicide is a deterrent, rather than actually viewing them as perpetrating a crime. Sometimes a prison cell with someone watching them is safer for someone with a tendency to harm themselves. Very rarely do authorities actually prosecute people for suicide. Even if they do, the goal is to get them in a shelter or get them counseling or get them medical treatment so they can get back to a healthy happy state of mind. These law also intends to protect loved ones. While we may not have a right or ownership over their lives, we do have some rights over our loved ones. We don't want to lose them to unfortunate circumstances and suicide laws try to serve this purpose. I think it is very easy to look down on people who commit suicide. Death actually is one of the biggest human fears. These people have been pushed so far that they prefer one of the biggest fears to everything else. It is not a pleasant place to be. Rather than blame them or ridicule them, the decent thing to do is figure out how we could have prevented it.
On a side not I believe "attempted suicide" is a prosecutable crime not "suicide". After successful execution of "suicide" there is no criminal alive to prosecute unless posthumous prosecution is an option.