DTM that halts on a minimum one input
Hint: Use the technique of dovetailing, mentioned in several of the comments. The idea is to maintain a growing list of simulated Turing machines, all of them copies $T_1,T_2,\ldots$ of the input Turing machine $T$, operating as follows:


Initialize $T_1$ with the $1$st possible input, and run $T_1$ for one step.
Initialize $T_2$ with the $2$nd possible input, and run $T_1$ and $T_2$ for one step each.
Initialize $T_3$ with the $3$rd possible input, and run $T_1,T_2,T_3$ for one step each.
And so on.


You get the idea. It is perhaps not obvious how to maintain the states of several Turing machines at once, but you can figure it out on your own.