Edit: Why was the only informative comment removed? It shared valid information that was exactly what I asked for. Glad I save the link before it got deleted.
Presumably you are referring to the cost of the war to the Socialist Republic of Vietnam (i.e., North Vietnam), not the Republic of Vietnam (i.e., South Vietnam).
In 1998, the CIA declassified, partially, a report from 1968 in which they, along with other government departments, assess the effect of the war on communist countries. It is divided into different segments, but the most pertinent for your question can be thought of as two big blocks: the first, running through page 31 of the report, is the indirect cost to North Vietnam's allies in the communist world, and the second, starting at the top of page 31, is the direct cost to North Vietnam itself.
The report goes through its methodology fairly comprehensively, and while it is certainly not perfect, these numbers can give you a general sense of what is otherwise a difficult nut to crack. I will break out some highlights here.
The report states the following estimates:
Indirect Costs to the Communist Nations
The USSR, China, and Eastern European communist countries sent around $1 - 1.5 B in military aid to North Vietnam (the equivalent of about $7.25 - $9 B in today's dollars) from early 1965 through 1968.
Additionally, these countries sent $1 B in economic aid to North Vietnam (the equivalent of about $7.25 B in today's dollars).
Logistical outlays, e.g., shipping costs, transferring and stationing soldiers and other personnel in Vietnam, and the "maintenance of North Vietnamese students" (i.e. indoctrination into communism, etc.), totaled about $150 MM, or about $1 B in today's dollars.
Military spending in the USSR went up by at least 3.5 BB rubles YoY from 1965-1967, suggesting more money was being spent to funnel troops and supplies to Vietnam, though no direct connection could be established.
The Vietnam War affected what had been softening trade restrictions between the USSR and the West, greatly reducing communist countries' ability to get advanced technology through trade, and hurting their economies due to the absence thereof.
Direct Costs to North Vietnam
The report states that the opportunity cost for the North in terms of conscripting large amounts of its young, male population to the military and to repairing damaged infrastructure and facilities due to the bombings is difficult to calculate but definitely impactful on North Vietnam's economy and future likelihood to be dependent on communist countries for foreign aid.
The report divides damage to North Vietnam's facilities into "Economic and Military" categories, and estimates that Economic damage is about $305.2 MM USD (about $2.2 BB in today's money), and Military damage is about $136.6 MM (or just under $1 B in today's money).
The report details damage to North Vietnam's industrial sector (which it says is small), stating that "a large part is out of operation" due to the bombing, and that there has been about $60 MM (about $422 MM in today's money) in damage to it (which it breaks out into line items like power plants and shipyards). It states that most of this damage was inflicted during the first half of 1967, and a decrease in frequency of bombings has allowed the North to restore some operations in these industrial facilities. It states that the bombings of shipyards did not significantly hinder production of boats.
Transportation infrastructure was damaged or destroyed to the effect of about $130 MM USD (about $950 MM in today's money), greatly reducing North Vietnam's ability to easily transport goods and people, and thus increasing those costs.
Though not directly targeted, agriculture and fishing were disrupted by the bombings to the tune of about $73.3 MM ($515 MM in today's dollars) in lost gains.
Exports were reduced as a result of the war, reducing North Vietnam's gains by about $41.1 MM ($289 MM in today's dollars).
Direct damage to fixed military facilities, such as supply depots and air defense systems, totaled about $75 MM (about $527 MM in today's dollars), while damage to military equipment (mostly aircraft) totaled $61.9 MM (about $400 MM in today's dollars).
The human cost of the war -- not discussed in this report, but estimated later -- to Vietnam is at least 1,000,000 soldiers, and 500,000 civilians (North and South), but likely far more than that, as the North Vietnamese government did not keep records, and has not shared what it does know.
For what it's worth, the CIA characterized these costs (except for the human cost, which they did not know of at the time), as relatively small to both the USSR and North Vietnam.
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