Banks were generally interested in cultivating an image of impregnability and absolute security for their clients. "Losing" money would be virtually unthinkable to a sane banker; word of mouth from any sufficiently lucrative incident would have people closing their accounts in short order, and a run on the bank could (and did) ruin establishments. Suddenly, our hypothetical banker has compromised his capital base, possibly incurred legal responses from his more influential victims and, by breaking faith, destroyed his credit with other lending and accounting institutions.

TL:DR Easily exposed fraud is rarely a good business practice.

Wealth, especially in Post-Norman England, had historically been held in land, and the rents collected from the same. The Domesday Book for example, was a systemic document prepared by William I in 1086 to account for the wealth of the landowners of England, so that they could be taxed.

From the Anglo Saxon Chronicle:

The king spent Christmas with his councillors at Gloucester, and held his court there for five days, which was followed by a three-day synod held by the archbishop and the clergy. At this synod Maurice was elected bishop of London and William bishop of Norfolk and Robert bishop of Cheshire: they were all chaplains of the king. After this the king had important deliberations and exhaustive discussions with his council about this land and how it was peopled, and with what sort of men. Then he sent his men all over England into every shire to ascertain how many hundreds of 'hides' of land there were in each shire. He also had it recorded how much land his archbishops had, and his diocesan bishops, his abbots and his earls, and--though I may be going into too great detail--and what or how much each man who was a landholder here in England had in land or live-stock, and how much money it was worth. So very thoroughly did he have the inquiry carried out that there was not a single 'hide,' not one virgate of land, not even--it is shameful to record it, but it did not seem shameful for him to do--not even one ox, nor one cow, nor one pig which escaped notice in his survey. And all the surveys were subsequently brought to him.

Here is a translated example of a typical entry:

The land of Robert Malet. Fredrebruge Hundred and half Glorestorp. Godwin, a freeman, held it. Two carucates of land in the time of king Edward. Then and afterwards 8 villains; now 3. Then and afterwards 3 bordars; now 5. At all times 3 serfs, and 30 acres of meadow. At all times 2 carucates in demesne. Then half a carucate of the men, and now. Woods for 8 swine, and 2 mills. Here are located 13 socmen, of 40 acres of land. When it was received there were 2 r.,' now 1. At all times 8 swine, then 20 sheep, and it is worth 60 shillings.

There is situated there, in addition, one berewick, as the manor of Heuseda. In the time of king Edward, 1 carucate of land; then and afterwards 7 villains, now 5. At all times 12 bordars, and 3 serfs, and 40 acres of meadow; 1 mill. Woods for 16 swine and 1 salt pond and a half Then 1 r., and now and 14 swine, 30 sheep, and 50 goats. In this berewick are located 3 socmen, of 10 acres of land, and it is worth 30 shillings. The two manors have 2 leagues in length and 4 firlongs in breadth. Whosoever is tenant there, returns 12 pence of the twenty shillings of geld.

Scerpham Hundred Culverstestun Edric held it in the time of king Edward. Two carucates of land. At all tomes there were 4 villains, and 1 bordar, and 4 serfs; 5 acres of meadow and two carucates in the demesne. Then and afterwards 1 carucate, now one-hal£ At all times 1 mill and one fish-pond. Here is located 1 socmen of the king, of 40 acres of land; which his predecessors held only as commended and he claims his land from the gift of the king. Then and afterwards there was one carucate, now 2 bovates, and 2 acres of meadow. At all times two r.[note: word indicated by "r" has not been identified] , and 4 geese; then 300 sheep, now 300 less 12; then 16 swine now 3. Then and afterwards it was worth 60 shillings, now 80; and there could be one plow. Walter of Caen holds it from Robert.

Heinstede Hundred. In Sasilingaham Edric, the predecessor of Robert Malet, held 2 sokes and a half, of 66 acres of land, now Walter holds them. Then 9 bordars, now 13. At all times 3 carucates and a half among all, and 3 acres of meadow, and the eighth part of a mill; and under these 1 soke of 6 acres of land. At all times half a carucate. Then it was worth 30 shillings, now it returns 50 shillings.

In Scotessa Ulcetel was tenant, a free man commended to Edric, in the time of king Edward of 30 acres of land. At that time 1 bordar, afterward and now 2. Then half a carucate, none afterward nor now. It was at all times worth 5 shillings and 4 pence; the same.

Source: The Anglo Saxon Chronicle

I assume you're asking about western banks here, but I'll answer with Japanese information anyway. The Japanese aristocracy, from at least the 8th century (but probably earlier, some scholars think the practice started in prehistoric times) kept their wealth, in whatever form, in storehouses. Here is a picture of one of the more famous storehouses that housed part of the imperial treasures. Pretty much any form of movable property could be held in these, and there are a number of ways architects worked to make them fire, pest, and moisture resistant. Temples, nobles, merchants, and agriculturalists all used similar (albeit smaller) storehouses.

In addition to movable property, a lot of wealth was held in land, and securing it from stewards just "losing it" or willfully stealing it actually caused a lot of problems through the medieval period. From what I've read, the vast majority of official documents coming out of the Kamakura warrior government (1185-1333) were about land rights, securing land tenure, telling land stewards to fulfill their obligations to absentee landlords (aristocrats who lived in Kyoto), or otherwise settling disputes between people with interest in the products of the land. The products being the movable income that either was consumed in some form (grain, fish, timber, fibers) or put in permanent storage (artisan products for the most part).

Aside from bugs, rats, moisture (it's really humid in Japan..), and fires (they happened pretty frequently), another threat to all of the wealth inside storehouses were thieves. From what I can tell, the only protection was the walls of whatever compound these were in. They don't seem to have been locked, and sometimes they're built as a part of a compound's wall, to keep them away from the buildings that may set fire, which means there's a plaster wall between a thief and treasure. So of course thieves devised saws to get through the walls. Still, they worked fairly well, which is why so many documents and artifacts survived until the age of museums and conservation.

One of the better sources for this that I've found is The Japanese Storehouse by Marc Treib.