Binary classification for imbalanced distribution of target/response class for age I'm trying to build/train model that depends on many attributes where age is the most important one (it has significant impact on AUC).
Overall target class count is quite balanced (+40% vs. -60%) and whole dataset is small (~150 samples).
This is a plot of target class (diagnosis) distribution for age:

Maximum frequency/count peaks are shifted in opposite directions for each target class (diagnosis)
I would like to increase general accuracy of model (e.g. GBM or GLM).
I wonder what I can do to minimalize impact of such distribution for predictions (false negative for young and false positive for old patients).
Are there specific methods for age adjustment, sampling or cost/penalty functions that increase performance (e.g. AUC) of model (especially for GBM or GLM)?
 A: Do not do anything to your sampling distribution. This is not a situation where we have strong class imbalance; there is no reason to perplex things.
Do assess the classifier performance based on calibration plots. In general what we want is a well-calibrated model. I would suggest trying a Generalised Additive Model (GAM) so non-linear relations between the predictor variables and the response can be taken into account.
Aside GAMs using a penalisation approach like elastic net, might be a good idea especially given the relatively small size available.
To that extent, 150 samples do not offer a lot room of generalisation so it would be essentially to cross-validate your results. The methodology presented in Beleites et al. (2013) Sample size planning for classification models is a good starting point.
Side-note for GBMs:  A GBM while great, usually does not offer well-calibrated probabilities out of the box. The extra calibration step (i.e. Platt scaling, isotonic regression, beta calibration, etc.) requires a hold-out sample and with 150 samples to begin with, this is too expensive at this point. It is more prudent to focus on learners like GLM/GAM that are well-calibrated out-of-the-box (see the CV.SE thread on Why is logistic regression well calibrated, and how to ruin its calibration?, for more details).
