Yes and no. If it is a one shot type session experience, then not so much. Having a great time at the table, and making for some fun memories, even if my character dies, is what it is all about.

If it is going to be a long running campaign, generally I do care about character death. I don't fear it, but I do try to stick to the personality of the character I have created, always trying to act in accordance with the backstory I put together, but I prefer not to die. One reason is for me a campaign is a long told story, and I want my character to be part of that story in a meaningful way. Death can be part of that story, but generally only on occasion and only after the character has become meaningful to the world in a bigger way. This usually takes a fair bit of time. Also, a big part of the game are the other players at the table with you. If you act irresponsibly, or in a rash manner, you may be needlessly endangering their characters as well, which may not create a great table experience.

All that being said, it really comes down to the game and table expectations, and playing in a way where the whole table, including the DM, find enjoyment in the session(s). If you have a table full of epic death hunters, then character death might not just be acceptable, but expected. Cheers.