It seems like drivers have been increasingly commenting that since cyclists don’t pay the gas tax or pay for licensing their bicycles they shouldn’t be using the road at all; cyclists don’t “pay their fair share” in other words. Well then, how are our roadways (interstates, highways, and city streets) are funded? Roadway funding may differ slightly from city to city and region to region but generally the funding sources are similar across the country.
Let’s look at the funding of highway construction and maintenance first – from the “gas tax.” This “gas tax” does not come solely from a tax on gasoline. Of the total state/federal funding for highways in the western states, 33% of the total funding for highway construction and maintenance “gas tax” ($.0193/gallon of the total $0.583/gallon) comes from sources other than a direct tax on gasoline purchases; sales tax, gross receipts, oil inspection fees, county and local taxes, etc. contribute to the federal highway dollars distributed to each region.  
Specific to WA State, when someone talks about fuel, or gas tax, ask them what kind of fuel. The gas tax is collected on not only motor vehicle fuel, but aircraft fuel, heating oil, natural gas, diesel and methanol  to name a few. If you fly, own a gas-powered lawn mower, motorcycle, off-road vehicle, etc. (some of which are subject to licensing fees which directly go to roadway construction and maintenance) you pay gas tax which directly pays for construction and maintenance of city, county and state roadways, the ferries, purchases right of way, traffic lights and signal maintenance, policing, and span bridge operations. 
Many purchasers of fuel in WA State are exempt or can take tax deductions from paying a fuel tax: off-highway vehicles (farm use), non-motive power uses (like generators), any motor vehicle belonging to Seattle Department of Transportation, or motor vehicles belonging to counties or municipalities, emergency vehicles, and more. Tribal purchases of gas are refundable up to 75% of the total gas tax paid.  I hardly see those who rally against cyclists using roadways because they don’t pay a “gas tax” telling the fire department to “get off the road” because they don’t pay a fuel tax.
Most of the Seattle roadway funding comes from the General Fund.  Transportation receives 5% of this fund which is collected from commercial parking taxes, employee hours tax, grants, local fees, property taxes (~37.8% of total property taxes go to cities/towns, counties, and “other” which includes the General Fund) , some federal funding and levies like the Bridging the Gap levy  that was passed by Seattle voters in 2006 which includes a mandate that requires at least 18% of the levy funds be spent on pedestrian and bicycle infrastructure.
75% of WA State roadways are paid for by state and federal gas taxes (of which it’s been established that not all of the federal gas tax is funded solely by a direct gas tax) and the rest from vehicle license fees, bonds and ferry fares, and other funding sources  . Highways are funded at about 57% by direct “user fees” or fuel taxes and vehicle licensing.  So how much roadways space is actually given to cyclists, considering that approximately 20%-30% of the direct funding of roadways and highways is funded by non-gas tax related taxes?*
The Seattle Sidewalks Initiative of March 2007 indicated that there was 3,946 miles of roadways within the Seattle city limits.** The 2009 Seattle Bicycle Master Plan lists a total of 67.6 miles of roadways with bicycle infrastructure (including 25.5 miles of bike lanes and 0.3 miles of Sharrowed “Shared Lane Pavement Markings” roadways); so the city officially allocates only 1.7% of the roadways to direct bicycle infrastructure and 0.65% for bicycle only use although 20%-30% of roadways are paid for by non-gas tax related sources.
Those who think that that they “pay for” the use of the roadway solely because they pay a license fee or gas tax are sorely mistaken. The average life of a roadway is 40 years and maintenance never ends once a roadway is constructed.  In order for a roadway to be truly paid for, revenues must meet the cost. The gas tax only pays for a small portion of the roadway and the true costs are subsidized by other taxes that come from a variety of sources.***
Long story short, unless a cyclist lives entirely “off the grid” paying no rent, no property or income tax, pays no sales tax, does not operate or pay for the use of anything that uses any type of fuel (airplane, ferry, chainsaw), that cyclist is helping pay everyone’s way.^
* While the federal government does not ban cyclists from using highways and freeways States can ban cyclists due to safety concerns. So although cyclists are paying for the construction and maintenance of highways and freeways, in many cases, those same cyclists cannot use them. 20%-30% is calculated by estimating that 25% of road construction and maintenance costs are collected from non-gas tax related “taxes and fees” and that approximately 33% of the federal gas tax funding distributed to the states comes from non-gas tax related sources.
**Note that the Seattle Sidewalk Initiative identified 598 miles of roadway, or 33% of all roadways within the Seattle city limits lack any sort of sidewalk. However there are actually 3,946 lane miles of 12-foot wide travel laned streets in Seattle city-wide . I have not discussed pedestrian use of sidewalks in this analysis as it is concerned with the roadway as used by vehicles rather than pedestrian usage.
***With the deterioration of the roadways today, combined with the use of heavier vehicles (SUVs) and the use of more fuel efficient vehicles, including hybrids, the fuel taxes actually collected are decreasing. Alternative road funding sources are being sought after by States and Cities. Nationally, according to StateLine.org, in 2006 alone, gas tax revenues were predicted to fall short by $1 trillion.     . WADOT recognizes the budgeting shortfalls due to the current economy as well. 
^”Since most communities have well-developed roadway systems that easily satisfy basic access, the need to increase roadway capacity usually results from motor vehicles’ relatively large space requirements. Even pedestrian and bicycle facility costs could be charged to driving if motor vehicle traffic is considered to degrade bicycling and walking environments, creating the need for separate facilities. This implies that most current road expenditures are the responsibility of motor vehicle users.”