While it's always wise to be suspicious, it would be silly to miss out on a genuine offer.
The 'Tell Ted' website is here:
http://www.eurostarpromotions.com/bookers_retu rn/email.html
The site looks genuine enough, although the web address is 'eurostarpromotions.com' and not 'eurostar.com' (which is the main address for Eurostar.
If you've got any doubts about whether that site is really a part of Eurostar's own site, all you need do is to phone Eurostar's customer relations department on 01777 77 78 78 or use the other means of contact given here:
http://www.eurostar.com/UK/uk/leisure/customer _care/questions_answers.jsp
Some hotels do ask guests to provide credit card details (so that extras such as drinks from the mini-bar can be charged for) even though they're travelling on business and their company has pre-paid the room bill. If Eurostar Customer Relations tell you that 'Tell Ted' is a genuine promotion (and can confirm that you're a winner), it should be safe to provide your card details.
Chris