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Have you considered a remote controlled switch on the power cord? You could place it between the wall socket and the device. I wouldn't recommend you build one yourself since you'd be messing with mains voltage. But there are many existing products in this field.
And while on the topic of soldering, the reason for one of the channels to be dropping out is likely to be because of a broken solder joint. Could even be a dry solder in the socket you're plugging into :)
Can you update your question with a schematic of the circuit as you have it at the moment? It's a bit difficult to imagine how you have things hooked up at the moment.
I forgot that RS232 has logic levels plus or minus 3 to 15 volts. I was thinking TTL. Are you aware that logic level one (enabled) will be -v? The server Tx will be either -v or +v. Also, 0 volts is not a valid logic level in RS232.
I was thinking more like a 7408 ic. I don't know why you'd use an opto triac? An opto isolator like a LiteOn LTV-816 might work. These are just ideas, I'm not sure how they'd go with the digital signals.
I might be opening my Dremel on the weekend to fix a problem with it running rough. It's a 230v model so I could take a look at the driver board and post the differences.
@Handshaking I'm not sure this can be solved with hardware alone. I'm assuming your modem can auto answer and enters data mode once a connection is established. Why isn't the software on the server waiting for the modem to be ready before sending data? Also, are you certain the modem can't be configured to ignore this data itself (ie, always be in data mode)? A hardware solution to your problem might be as simple as an AND gate between the modem DCD, server TX to the RX on the modem.