We learn from history that we learn nothing from history.
--George Bernard Shaw
If it can be done, it will be done, no matter the consequences.
Lets take computer viruses as an example of a technology with 'dubious benefits'. There are currently over 50,000 computer viruses and that number is growing rapidly. Each of these viruses has been handmade by a programmer, a person with good education, intelligence, dedication and more than average knowledge about computer systems, automation and programming. After developing these malicious programs called virus, they get spread intentionally.
One might think that if human viruses could be man-made they too will be developed, tested and distributed.
This might be truth for any development, invention and product of science and technology. If it can be developed, it will be developed, tested and used.
Following the above way of thinking, If time travel and sending messages to the past is possible and are really going to be invented, then it will be developed, tested and used.
If Time travel really could exist it would be a dubious technology, as it could unintentionally change the past.
Time travel is one of those unique sciences in the sense that if it gets invented, the consequence must be with us already. The effects of such a technology, although, possibly, clearly visible, might not be recognised. Potential time travel related events might have previously accepted alternative explanations and interpretations.