Does having multiple steamworks actually help?

Well Steamworks are effectively useless when you haven't researched/archived the adequate upgrades, but does several Steamworks multiply the production of each upgrade?

Yes. Every single benefit Steamworks are capable of is benefited by having more. Specifically:

• Printing Press: The rate of manuscript production is equal to 0.0005 manuscripts per steamworks per tick (The code has a comment that rates this as "2 per year per SW"
• Factory Automation: The amount of resources produced when this activates is directly based on how many Steamworks you own.
• Magnetos: The Magneto building you can get later increases your global production rate by 2% for each magneto. This effect is boosted by 15% for each Steamworks (So, with four Steamworks, each magneto is giving 2 * (1 + 4 * .15) = 3.2% production boost)

There is, however, one caveat when making additional Steamworks: If you build enough steamworks so that you are just barely unable to get, say, your wood levels back to full in a year after the Automation activates, but you are able to get minerals back to full, then you run into a minor problem: The automation will trigger at the start of the year, because you had enough of at least one resource, and it won't be until the next year before it makes Beams again.

If you plan to idle for a long time, it can thus be useful to make sure you're getting all the resources you can currently convert back to full before the Steamworks fires off again. Or, you can get the Advanced Automation upgrade; it allows the Steamworks to automate twice per year, which helps offset this problem.