Keyboard and mouse wired to computer from far away
You could get headphones if noise is the issue.
Some alternatives:
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Extenders, repeaters, switches
That's probably not the ideal nor the cheapest solution, but it's probably the easiest considering you already have the cables and the peripherals. Google for those for each cable type and you'll get a bunch of results.
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The network
I think this is the best solution. If you have a laptop or another computer or even a smartphone you might be able to remotely access your old computer on the basement not only from the 2nd floor but from anywhere - via the internet. You could also create a LAN, but that involves buying switches and/or Ethernet cables - see below to max length. You could even buy a wireless router and create a WLAN.
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Wireless peripherals
You could buy a wireless mouse and keyboard, but sending the video to the display would be a bit tricky and involve either the buying of expensive equipment or setting up a WLAN. Not to mention the signal loss that would certainly be present between the 2nd floor and the basement.
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Good ol' cables
- USB cables: Limit appears to be around 5m so unless you can dig a hole on the floor it's probably no good.
- Old mouse and keyboard cables (PS/2 ports): Never heard of a limit for those, doing some googling it seems to vary a lot from one model to another.
- VGA cables: It also depends on the quality of the cable, and apparently on the screen's resolution - more or less data being transferred. Doing some googling there are reports of people using 15m long cables with no signal loss.
- Ethernet cables: Very high limits - 100m it appears - I have about 30m of cable connecting my computer to the router and no speed is lost. You'd need to set up a LAN, though, i.e. you'll need a second device.
You are looking for a "VGA, keyboard, mouse extender" - there are many models, with prices ranging around 50-100 USD. Some models allow one V/K/M to control more than one server, in case you need it.
Gadgets such as this - but you can find them on Amazon too (and elsewhere). Some require just an ordinary CAT5 cable to be pulled, and send everything through that. There are also (but more expensive) units that use WiFi, or any Ethernet carrier, which means you may try and run them over a PowerLine adapter using your domestic power cables as network (i.e.: you plug a PowerLine in a power outlet in the cellar, plug the extender on the PowerLine, plug the other PowerLine in a power outlet on the second floor, with the second half of the extender plugged in. The two PowerLines, if they're attached at the same electric meter, establish a 85, 100, or 200 Mbps connection [depends on models, wire quality, and radio noise in the neighbourhood]. The two extenders "see" an Ethernet connection and route video/mouse/keyboard signals through that, and Bob's your uncle.
Or, the ultimate "extender": you run a Remote Access server on the server, and run the client on a laptop computer or thin client (e.g. a $80 Raspberry mini computer). Connect the two machines with an Ethernet cable or WiFi connection. The Raspberry in thin client mode sucks so little power it doesn't need fans, and it boots from a solid-state SD card. Plug VGA/HDMI, mouse and keyboard, and you have a completely noiseless setup.
Another possibility is to enclose the noisome computer in a foam box, with a silenced air duct for ventilation. There are already-made foam boxes that claim noise reduction up to 30 dB, but you can build one for about $30 ($50 if you also add low-freq damping feet, e.g. Dynavox DH-919 or Alu-Spike).