Does the leaves of papaya plant help in increasing platelets in a case of dengue?
So With some digging, I found some articles/presentations that provide evidence that fresh papaya leaf extract does increase platelet counts. The first link tested this in a rodent model and found that in the test group treated with the leaf extract, there were significant increase in platelet count and red blood cells (RBCs) from the control group. Here is the summary of their results trend from the paper:


Additionally, I found a Multi-centric, Double-blind, Placebo-controlled, Randomized, Prospective Study and a meta-analyses that supports the observation that papaya leaf extract does in fact show statistically significant increase in platelet count than that of the control groups. (Dengue fever causes platelet counts to plummet). [#2-3]
Below is the image of platelet counts from the Multi-centric, Double-blind, Placebo-controlled, Randomized, Prospective Study (2):

Fig. 1: Comparison of platelet count between two groups during different
timepoint


A good criticism of the meta-analyses on Papaya leaf extract is that something being statistically significant does not equate to clinically significant. But, the placebo-controlled, double-blind prospective clinical trial provides some evidence that in this case, it is actually clinically significant. 

However, the mechanism by which this occurs is not well understood. According to the 4th article, "The secondary metabolite of plants origin makes up a vast repository of compounds with a wide range of biological activities". The presentation (5) states some possible mechanisms by which the extract can inhibit the Dengue virus/vector. 


Does Carica papaya leaf-extract increase the platelet count? An experimental study in a murine model
Efficacy and safety of Carica papaya leaf extract in the dengue: A systematic review and meta-analysis
A Multi-centric, Double-blind, Placebo-controlled, Randomized, Prospective Study to Evaluate the Efficacy and Safety of Carica papaya Leaf Extract, as Empirical Therapy for Thrombocytopenia associated with Dengue Fever.
Dengue fever treatment with Carica papaya leaves extracts
Role of Papaya in Dengue fever, an evidence based Answer