You don't want the weight placed by a search engine to carry over to the new website because along with it, will be transferred all the negative weight from low quality / spammy content/links etc.
In order to just refer legitimate traffic to the new website, you'd need to implement a 301 redirect and exclude search engine crawlers such as
GoogleBot so that they do not get redirected to the new website. (So you'd be redirecting humans and not bots). You'd have to be wary of potential cloaking risks though.
Alternatively, could you not just display a notification or new page at your old website advising all visitors of the new website and ask them to visit that one instead?