You have two risks from buying on the first day:
1) there could be a fundamental issue of some sort with the phone. Something along lines of how some people perceived the old antennagate or bendgate "scandals" (I'm not looking to refight those here). Assuming you traded in or sold your old phone before realizing the problem, you might be stuck with a new phone that you don't like as well as your old phone. This kind of risk is probably lower with the S models.
2) there could be a bad batch of parts or a flawed assembly technique in the first batch that wasn't discovered until mass production. This is probably something that could be fixed or replaced under warranty, so is less of a concern. This is probably equally (un)likely in the S models and the base models. It is definitely less likely with phones in later production runs.
If you are willing to switch systems back-and-forth between phones, you would have the risk that another phone, Android or Windows, might come out shortly after your purchase that you would like better. But I don't think there are many people like that, especially among iPhone buyers. Our only choice seems to be big screen or small screen!
You have to decide how likely these problems are. I think the first and second problems are more likely with the company such as Samsung that produces many models each year than with Apple. My response to this was to pre-order my iPhone 6s Plus for the first day. And I've chosen similarly in the past. I figure that if this is the phone I want, I want to start enjoying it as soon as possible! I am happy with my choice so far.
If something comes up you have a year to get it fixed or replaced (two with AppleCare).
