There are some plausible arguments for having "if $a$ then $b$" true when $a$ is false (like suggested ex falso quodlibet). But the fact is $\rightarrow$ doesn't even try to capture the if-then relation between propositions. $a \rightarrow b$ is defined as $\neg a \vee b$, and it's obvious why that's true when $a$ is false.
The actual if-then relation can be more appropriately captured by, for example, $a \Rightarrow b$. This is not propositional logic statement (rather metalogical), it says "it's impossible for $a$ to be true when $b$ is false".
Or better yet, use modal logics with modalities of necessity (physical, metaphysical, logical etc.): $\square (a \rightarrow b)$. This is much closer to capturing if-then relation of everyday use. Interpretation is "it's (physically/metaphysically/logically/...) impossible that $a$ is, but $b$ isn't". In fact trying to formalize if-then was perhaps the main reason why alethic modal logic was invented in the first place.