Snopes said no:
But the Slate article (presented as a question in its title) simply strung together circumstantial details to suggest Trump had a server connection to Russia. A concurrent and a subsequent look at its conclusions (the latter by a cybersecurity expert who was not anonymous) asserted that the claims were unsubstantiated and likely amounted to nothing.
The specific circumstances were checked by Robert Graham of Errata Security. One snippet of that:
When you view this "secret" server in context, surrounded by the other email servers operated by Listrak on behalf of Cendyn, it becomes more obvious what's going on ... It's Cendyn that registered and who controls the trump-email.com domain, as seen in the WHOIS information. That the Trump Organization is the registrant, but not the admin, demonstrates that Trump doesn't have direct control over it ... When the domain information was changed last September 23, it was Cendyn who did the change, not the Trump Organization. This link lists a bunch of other hotel-related domains that Cendyn likewise controls, some Trump related, some related to Trump's hotel competitors, like Hyatt and Sheraton.
Note that the trump-email.com server was replaced with trump1.contact-client.com which is owned by Cendyn.
Anyway, the evidence is that Cendyn manages marketing campaigns for hotels. Alfa Bank has used the Trump hotels. The server that Cendyn manages for Trump hotel marketing sent them some follow-up emails. The bank did some kind of follow-up, either automatically or manually (or both).