One of the key considerations here is how much time the appliance will spend turned on. If an old CRT TV is hardly used, then the environmental footprint of continuing to use it is very low, even if it uses 5 times as much electricity as an LCD one.
Secondly, the way the electricity is generated makes a substantial difference. If renewable energy is used to run the old TV, then the environmental footprint is much lower.
The energy used to manufacture a TV (or any electronic device) is substantial and also needs to be considered. I would expect energy used for most electronics to have come from non-renewable sources. To alleviate this environmental cost, it could be an option to purchase a second hand TV rather than a new one. A second hand one might be for sale because someone is purchasing a larger model.