While you can’t style select dropdowns fully with CSS alone, you can improve their appearance with some basic styling.
Source Code
select {
background-color: white;
border: thin solid blue;
border-radius: 4px;
display: inline-block;
font: inherit;
line-height: 1.5em;
padding: 0.5em 3.5em 0.5em 1em;
margin: 0;
box-sizing: border-box;
-webkit-appearance: none;
-moz-appearance: none;
appearance: none;
background-image: linear-gradient(45deg, transparent 50%, gray 50%), linear-gradient(135deg, gray 50%, transparent 50%);
background-position: calc(100% - 20px) calc(1em + 2px), calc(100% - 15px) calc(1em + 2px), 100% 0;
background-size: 5px 5px, 5px 5px;
background-repeat: no-repeat;
}