How robust is the normality assumption for two sample independent t-tests? I'm working with a dataset where the distributions are non-normal.

They are independent samples where the variances are roughly equal.
Can I still run a two sample independent t-test? If not, is there an alternative way to test whether the mean for these two samples is significantly different? 
Also, I know I can normalize with a BoxCoxTransformation. Does that work for the case of t-tests?
 A: *

*When you ask about the robustness of the t-test it's important to consider whether you're interested in the level-robustness or what happens to power.
The t-test is somewhat level-robust -- i.e. its significance level is not terribly affected by mild deviations from normality. However, it is not so robust to very strong skewness. 
The t-test is less robust when it comes to power, and once you move even a little away from the normal, it is often beaten by reasonable alternatives. Make the tail just a little heavier for example, and the Wilcoxon-Mann-Whitney has better power under shift alternatives.

*
I know I can normalize with a BoxCoxTransformation. Does that work for the case of t-tests?

Well, yes, but you need to do more work as far as assumptions and interpretations go; you're testing for a mean-shift on the transformed scale. It still works as a test of identical distributions under the null vs a particular form of stochastic dominance (greater than/less than) under the alternative.
I would not suggest searching for a transformation; you have what's essentially a scale shift. If you want to transform, take logs (which convert that scale shift to a location shift). That wouldn't be a terrible candidate for a t-test, even if there's some skewness remaining (e.g. you might get some left-skewness after transforming, but it won't be of huge consequence in this case).

*Nonparametric tests are only going less powerful when the assumptions underlying the test are actually true -- and in many cases only a little or sometimes not at all. So for example, if you choose a slightly heavier-tailed distribution than the normal then under a location shift alternative in many cases the t-test is less powerful than the Mann-Whitney (though there are other nonparametric tests that match the small-effect-size power of the t-test even at the normal, in large samples)

*However, in this case there's really no need to test, you can tell by mere inspection -- the scale of the second sample is about twice the scale of the first and your sample sizes look to be in the vicinity of 8000. Any reasonable test for a scale shift would find such a huge effect (a Wilcoxon-Mann-Whitney would do fine, as would a t-test on the logs, or a Gamma GLM) and would easily find it at any reasonable significance level. Your p-value is going to be vanishingly small in every one of those cases; power is largely a non issue with such a large effect size.

*If you must test in spite of the fact that it's so obviously significant by eye, I'd suggest a gamma glm:
Here's some samples that look a lot like yours:

Here's the relevant test of a difference in scale using a gamma GLM:
summary(glm(values~ind,data=xy,family=Gamma(link="log")))

[...]
    Coefficients:
                Estimate Std. Error t value Pr(>|t|)    
    (Intercept)  3.82282    0.01572  243.13   <2e-16 
    indTRUE     -0.68011    0.02224  -30.59   <2e-16 
    ---
    (Dispersion parameter for Gamma family taken to be 1.977846)

        Null deviance: 42845  on 15999  degrees of freedom
    Residual deviance: 41030  on 15998  degrees of freedom
    AIC: 136983

Specifically, it's the test of the coefficient of indTRUE that tells the story (though in this case, with that as the only predictor, you can also base the test on the scaled change in deviance (1815/1.9778=917.7) by comparing it to a chi-squared 1. This would correspond to a two-tailed z-test with a statistic of $\sqrt{917.7}\approx 30.3$ (it's no accident that this is essentially the same as absolute value of the statistic for the coefficient we started with).
Of particular interest here is that the estimate of the ratio of scales (i.e. $\text{scale}_\text{TRUE}/\text{scale}_\text{FALSE}$) is  $\exp(-0.68011)\approx 0.507$ (since we used the log-link), which is very close to the ratio of the population values in the data-generation (as it should be with such a large sample size).
By comparison the t-test on the logs also easily identifies that there was a change of scale (hence a location-shift in the logs), but you must be a bit more careful in estimating its size in this case (though the interval for it is straightforward). [The above data are not lognormal, by the way. It won't really matter for this specific situation.]
With a Wilcoxon-Mann-Whitney there's no need to take logs as far as the p-value goes (it makes no difference, and works the same for any monotonic transformation), but if you're trying to use it to place a confidence interval on the size of the scale-shift (whose log is the location-shift on the log-scale) would be the one to use. On my data the 95% interval for the ratio of scales is (0.477, 0.534), neatly bracketing the true value (I used 0.5).
A: It looks like you have large samples here for both the Response=True population and the Response=False sample, so the assumption of normality for testing for the difference in means from two populations can be greatly relaxed.  The Central Limit Theorem kicks in for large samples and assures us that the sample means of the two populations are Normally distributed.  So you can simply carry out the independent samples t-test without checking to see if the parent distributions are normal.
