Proof via induction for small-step semantics
The induction should be on the derivation of $t \to^* t'$:


If $t \to t'$ in one step, then we get the desired result by the E-IF rule.
Suppose $t \to^* t'$ because $t \to t'' \to^* t'$. By the E-IF rule we have
$$(\mathtt{if}\;t\;\mathtt{then}\;t_2\;\mathtt{else}\;t_3)
    \to
    (\mathtt{if}\;t''\;\mathtt{then}\;t_2\;\mathtt{else}\;t_3)$$
and by the induction hypothesis
$$(\mathtt{if}\;t''\;\mathtt{then}\;t_2\;\mathtt{else}\;t_3)
    \to^*
    (\mathtt{if}\;t'\;\mathtt{then}\;t_2\;\mathtt{else}\;t_3).$$
These two together give us the desired
$$(\mathtt{if}\;t\;\mathtt{then}\;t_2\;\mathtt{else}\;t_3)
    \to^*
    (\mathtt{if}\;t'\;\mathtt{then}\;t_2\;\mathtt{else}\;t_3).$$