First, know that by definition liquid crystal molecules can be negatively or positively dielectric. Therefore, using the term negative birefringence is not a right thing to say. The birefringence is always positive, and it is the dielectric that can be either positive or negative. Most discotic liquid crystal molecules are negatively dielectric.  

There are couple of ways you can verify that:

1- You can draw the molecule structure in software like "chem draw" to see if the molecule is negatively or positively dielectric.

2- Another thing you can do is you can fill a liquid crystal cell with planar alignment with your material, adjust the rubbing direction of your cell to be 45 degree with respect to the crossed polarizers, then apply ac electric field across the thickness of the cell, if at  higher voltages, under POM the image doesn't go dark between crossed polarizers, it means that the material is negatively dielectric, and in case it becomes fully dark, it is positively dielectric. 

Remember that some materials change their behavior based on the applied frequency.

Dear, to recognize between them you must be take in consideration the values of two refractive indices (ordinary and externally) and according to equation (n ord. - n ext.) select the negative and positive.

Dear Salili

I agree with your answer in case of dielectric cost . But birefringence depends up on whether  n_e more or n_o  more. So birefringence may not be positive always.

My interst is how can we say birefringence is positive or negative with out calculating n_e and n_o.

In my opinion, what you could do is to check how birefringence change by rotating analyser in cross polarizing microscopy. If positive your birifringence rotate as the analyser, otherwise it is negative. 

I am sorry but I have to correct myself that some discotic molecules do have a negative birefringence, and what is always positive is the retardance. That's why people make films from disc-shape molecules as the compensator for Displays and other applications. Therefore, the n_e which is normal to disc is smaller than the n_o in discotic molecules, and that's how delta n becomes negative.

Dear Tardani, are you sure about checking birefringence sign by rotating analyser.

To my knowledge, by rotating analyzer, birefringent color will change and finally at 90, we will have complete bright in color. But by rotating sample, we will know rubbing direction and whether it is birefringent or not. Let you please conform with your answer.

Dear Venkata, sorry, you are right. 

In a certain way, rotating sample or rotating the analyser are similar. I mean rotating the analyser can show how the domain directors are distributed in the sample (consider for instance the extreme cases of parallel or perpendicular directors compared to analyser). In this sense, rotating the sample will rotate these directors in a similar way. However, you are right because rotating the analyser reduces the overall birefrincence of the sample making hard to follow directors motion.

Hope this could clarify. I am sorry for the misunderstanding!

To determine which refractive index (no or ne) is higher you can try the following technique. You'll need homogeneously oriented planar sample and compensator like violet Lambda plate or quartz wedge (preferable due to wider birefringence range). The whole idea of the measurement is to compensate birefringence of your sample under crossed polarizers with precisely known birefringence of the compensator. Depending on the orientation of the compensator (with known optical axis direction) you'll see which refractive index is larger: no or ne.

Depending on the absolute magnitude of (ne-no) you should adjust cell thickness in order to match cell retardation H*(ne-no) with retardation range of your compensator.

This technique is used for LC characterization, for example here: http://journals.aps.org/pre/abstract/10.1103/PhysRevE.72.041711

Thank you Uliana. I am Sorry I didn't check

I think the measurements are two used .