There is no direct correlation between stability and percentage of beta strand and helix secondary structure. However, highly alpha-helical protein frequently are less problematic upon unfolding, proteins with high beta strand propensity are more likely to form amyloid-like aggregates.
See here for the regions of the ramachandran plot corresponding to the different secondary structures: http://www.cryst.bbk.ac.uk/PPS95/course/3_geometry/rama.html or simply consult wikipedia http://en.wikipedia.org/wiki/Ramachandran_plot
Silly but in some way interesting question? People struggled to understand protein structure for many years now and there still no consensus. My personal take it was expressed in my PNAS paper (2009) in which we stated that every protein is metastable and to reach its stability conditions it has its own design principle in which it combines in a unique proportion solid and liquid state characteristics. The nontrivial question would be which of the secondary structure is more "stable". Stability has two separate meanings. Stable means on one hand "solid" very rigid on the other hand it means well defined shape but repetitive nature of dynamical behavior. "LIquid" in this language would mean something not well repetitive and behaved without a specific structure.
My experience is that beta sheets have more twisting mobility but are relatively rigid while helices are like springs and are more dynamic. However, all my papers provide examples of both types of secondary structure providing basis for mobility necessary for function. On the other hand if you consider stability as temperature resistant then definitely both secondary structure elements have their own failings and the entire field of extremophiles is not well understood. I am not sure I helped you much but at least provided some basis for understanding the implications of your question that is remarkably complex upon closer examination. Good Luck.
Beta sheets are stabilized by hydrophobic contacts and backbone hydrogen bonding. Alpha helices are largely stabilized by backbone hydrogen bonding. That is, local interactions dominate in a helix, whereas a sheet is stabilized by long range contacts. So, a sheet is slightly inferior in terms of stability.
As mentioned, sheets have more hydrophobic contacts, mutation or misfolding, could result in exposure of hydrophobic residues. In which case, proteins aggregate to prevent hydrophobic exposure/avoid entropic penalty. folding aggregates, inclusion bodies and amyloid
This is also one of the primary reasons why designing beta sheet peptides is not easy, i.e. they can very easily aggregate and form amyloids : Page on sciencedirect.com
Both ordered and disordered aggregates show a high/higher beta sheet content. So, the energy barrier for helix---> sheet transition is higher.
Beta sheets induce the stability of protein structure. you can see the following papers. I highlighted the most important papers. these structures connected by a network of hydrogen bonds and occur widely in proteins. Although the importance of beta-sheets in the folded structures of proteins has long been recognized, there is a growing recognition of the importance of intermolecular vinteractions among beta-sheets. it seem these structures have great role in the stability and folding of proteins.
1-Protein Science (2004), 13:1134–1147**
2-ACCOUNTS OF CHEMICAL RESEARCH Vol. 41, No. 10 October 2008 1319-1330**