we can do it in two ways one is in informal way and another in formal way

(1) we have to check every length string in both language ,if there is some thing common then we can say yes ,but if we have nothing in common then we have to check every possible string forever

(2) we can use reduction to proove this.we all know PCP(post correspondance problem) is undecidable . somehow if we are able to reduce PCP to Empty Intersection problem then we can say that Empty Intersection problem also undecidable.

let's try in PCP we have to find some solution of given two sequence of strings .

for example A and B are two sequence ,A have set strings {1,10111,10} and B={ 111,10,0} .both have three strings name as 1,2,3 . we can use these string as many times to find solution. like 2113 is the solution of above sequences .

$A_2A_1A_1A_3$=10111 11 10

$B_2B_1B_1B_3$=10 111 111 0

now we can convert these sequences into grammars-

$S_A =1S_Aa|10111S_Ab|10S_Ac|1a|10111b|10c$

$S_B=111S_Ba|10S_Bb|0S_Bc|111a|10b|0c$

a,b,c repesent the record which string is concancated .

these two grammars generate languages have something common which also be solution of PCP and they have no common then no way to solve .so we have reduced PCP to Empty Intersection problem . so Empty Intersection problem also undecidable.