Quick guide into training highly imbalanced data sets I have a classification problem with approximately 1000 positive and 10000 negative samples in training set. So this data set is quite unbalanced. Plain random forest is just trying to mark all test samples as a majority class.

Some good answers about sub-sampling and weighted random forest are given here: What are the implications for training a Tree Ensemble with highly biased datasets?

Which classification methods besides RF can handle the problem in the best way?
Max Kuhn covers this well in Ch16 of Applied Predictive Modeling.     
As mentioned in the linked thread, imbalanced data is essentially a cost sensitive training problem. Thus any cost sensitive approach is applicable to imbalanced data.
There are a large number of such approaches. Not all implemented in R: C50, weighted SVMs are options. Jous-boost. Rusboost I think is only available as Matlab code. 
I don't use Weka, but believe it has a large number of cost sensitive classifiers. 
Handling imbalanced datasets: A review: Sotiris Kotsiantis, Dimitris Kanellopoulos, Panayiotis Pintelas'
On the Class Imbalance Problem: Xinjian Guo, Yilong Yin, Cailing Dong, Gongping Yang, Guangtong Zhou