I just had the same problem on OpenBSD 6.3 on a Intel Kaby Lake setup that I just installed, however I seem to have solved it. I was migrating from an Ivy Bridge setup that did not have that problem, though my mouse had a problem on the old setup, which is that it kept connecting and disconnecting before I started FVWM (which is the default window manager) ... either by logging in with startx or with xenodm. It is a blingy asian-made gaming mouse with weights and DPI settings and LEDs and all that stuff (I use it because I like the double-click button).
I chose to login with xenodm during installation. The keyboard did work for me after I logged in – I could use it to move the mouse around in FVWM with ctrl+shift+arrow-key. This allows one to move the cursor over the open xterm and start typing as focus-follows-mouse position in FVWM by default (you don't have to click in the xterm to start typing... I don't know a FVWM keyboard substitute for mouse clicks). I was using the same mouse, so I decided to try a different one... I plugged in an ancient microsoft mouse which worked immediately. Curiously after I plugged it in the blingy gaming mouse started working also. I rebooted, logged in as root and used "rcctl disable xenodm" as suggested. Logged into FVWM with startx and behold... the new mouse is working. Rebooted... it's still working.
tl;dr I tried a different mouse for a few minutes and now the original mouse is working
Moar: I just found out the mouse will work when plugged into a USB 2 port and exhibits its old bug of constantly connecting then disconnecting, but not when plugged into a USB 3 port unless I enable "XHCI Hand-off" in the BIOS. I don't know exactly what the issue between this Zelotes T-90 mouse and OpenBSD is... I suspect the mouse might be requesting a USB polling rate above the default 125Hz and OpenBSD can't support it.