There is no problem doing this and it can work just fine and be as secure as wifi can be. You made the first right step by running the ethernet cable. Have you checked the that cable works? That is have you plugged it into a LAN port on your router and a computer in downstairs and gotten internet access. That is where I would start then work on the WiFi.
Let's call the upstairs router the "router" and the downstairs router the "access point" or just AP.
You have internet at the router. Connect the ethernet cable to a LAN port on it. You will need to access the routers setup and find the settings for the IP range it assigns to clients through DHCP. Either change the range to free up a few IPs or select the highest number in the range.
Go to your AP and enter the setup page. If it has a setting AP mode use that. Otherwise turn off the DHCP and set the LAN IP to the number you selected from the router. Turn off the WiFi encryption. Plug the Ethernet into a LAN port. Save and restart the AP. Use the WiFi to see if the Internet is working. If it is, then reset the WiFi encryption. If not You may have to report back here for more advice.
You can use the same SSID on both or not. If you have equipment that allows multiple SSIDs you can set one SSID the same on both and each with a unique SSID too.
These are rather rudimentary instructions, but if you had enough savvy to run ethernet and attempt this on your own, you can probably work out the details for your particular equipment.
connect a computer via Ethernet from the 2nd WiFi.
You will need a Client Bridge for this. Some routers will function as a bridge and there are some bridge only specific equipment. It is sometimes cheaper to use a regular WiFi adapter on the computer though. But IME the bridge is more reliable.