To add snowflakes without using any images you just need to add a single line of JavaScript code. Follow the steps given below to add the snowfall effect in your blog.
Note: Before making any changes to your template, take a full backup of your blogger template.
#Step 1. Log in to your Blogger Dashboard and Navigate to Design> Edit HTML and Check theExpand Widget Templates Check box.

#Step 2. Find this code: </body> and put below JavaScript code right before the </body> tag.
<script src='https://sites.google.com/site/addinblogger/how-to-add-in-blogger/AddSnowfalleffectinblogger.js' type='text/javascript'></script>
#Step 3. Save template and done!.
Note: If background color of your blog is black then it will look very pretty, because of the snow color, which is of course would be white :)