I think the answer to your question is pretty well-addressed by Gerald in the comments.  Let me just throw in a couple of interesting points.

1)  Euler knew what he was doing.  He had a tremendous ability for mental calculation, and verified to his satisfaction that his sums converged.  He was well aware of the divergence of the harmonic series, so it's not like he was unaware of the fundamental issues surrounding "treating the infinite."  If a sequence converged in accuracy by one digit every couple of terms for 30 or 40 consecutive terms, this was good enough for him to be convinced.

2)  Even if Euler's proofs were not the most rigorous, that is not to say that modern mathematicians don't appreciate the methods behind them.  I'd say more often than not, the hard part of mathematics is figuring out what <i>should</i> be true.  Any student armed with Fourier analysis (and probably less) could re-derive many of Euler's formulas ($\zeta(2)$ in particular) -- few, however, would be able to play with the series even heuristically to figure out what the nontrivial contribution to the sum was, and fewer still would arrive at $\frac{\pi^2}{6}$ without prior exposure.  If Euler were to rediscover his results in today's academic atmosphere, I suspect he would be hailed for his great insight into what "should be happening," and have a very successful career providing graduate students amazing problems which needed details filled in.

3)  Even today, heuristics form an important part of mathematical research, so the legacy (if you will) of Euler's approach is still alive and well.  The Cohen-Lensta heuristics, as a more modern example (or maybe even analytic conjectures in general...maybe even something like BSD) might be considered as fundamental pieces of insight gleaned from heuristic reasoning and experimental data.