We have to divide between two cases
1.) The NA values of the response do not occur systematically:
Without a value of your target/response variable $Y$ any given row becomes useless for binary logistic regression or most other supervised classification algorithm. Supervised means that your examples must be labeled while unsupervised algorithms do not use the label information.
In your situation you can use those rows without the behavioural response in unsupervised preprocessing / feature generation techniques.
As an example you could take all given samples of your independent variables $X_1, \ldots, X_k$ to calculate principal component analysis (PCA) components $PC_1,\ldots,PC_k$. Then later you could train your bbinary logistic regression on a subset of those components $PC_1,\ldots, PC_l$ with $l \le k$.
Why is this a good idea? According to wikipedia
https://en.wikipedia.org/wiki/Principal_component_analysis
This transformation is defined in such a way that the first principal component has the largest possible variance (that is, accounts for as much of the variability in the data as possible), and each succeeding component in turn has the highest variance possible under the constraint that it is orthogonal to the preceding components.
So by taking the first principal components you maintain most of the variance in your samples but you reduce the number of variables and with it the chance of overfitting. Further the information contained in the response less samples is also included.
2.) The NA values of the response do not occur systematically
Here further analysis is necessary. You could try to train a classifier that predicts if the response is NA or not which results in the two classes $C_2=\{Y\neq NA \}$ versus $C_2=\{Y==NA \}$. If you can predict the NA response values than this is not a two classes problem $Y/N$ but a three classes classification taks $Y/N/NA$.